Simplex Method for Solving Linear Programming Problems with Fuzzy Numbers

Transcription

1 Word cde of Sciece Egieerig d Techoog Sipe Method for Sovig Lier Progrig Probes with Fuzz ubers S H sseri E rdi Yzdi d Zefri bstrct The fuzz set theor hs bee ppied i fieds such s opertios reserch cotro theor d geet scieces etc I prticur ppictio of this theor i decisio ig probes is ier progrig probes with fuzz ubers I this stud we preset ew ethod for sovig fuzz uber ier progrig probes b use of ier rig fuctio I fct our ethod is siir to sipe ethod tht ws used for sovig ier progrig probes i crisp eviroet before Kewords Fuzz uber ier progrig rig fuctio sipe ethod F I ITODUCTIO UZZY ier progrig first foruted b Zier [0] ecet these probes re cosidered i sever ids tht is it is possibe tht soe coefficiets of the probe i the obective fuctio techic coefficiets the right-hd side coefficiets or decisio ig vribes be fuzz uber [] [] [5] [6] [] [8] [9] I this wor we focus o the ier progrig probes with fuzz ubers i the obective fuctio Verdeg d et [] [9] proposed the equivet pretric ier progrig probes for these probes b use of certi ebership fuctio d proposed du ethod for fuzz uber ier progrig probes Here we first epi the cocept of the copriso of fuzz ubers b itroducig ier rig fuctio Moreover we describe bsic fesibe soutio for the FLP probes d stte optiit coditios for these probes Fi we provide soe iportt resuts for FLP probes d we propose sipe gorith for sovig these probes II DEFIITIOS D OTTIOS We first review ecessr bcgrouds of fuzz sets theor SH sseri Deprtet of Mthetic Scieces Shrif Uiversit of Techoog Tehr Ir (correspodig uthor: sseri@thshrifedu ) E rdi is with Deprtet of Coputer Egieerig Tr Uiversit Edire Ture (ebrurdi@tredutr) Yzdi Fcut of sic Scieces Mzdr Uiversit bosr Ir ( zdi@uzcir ) Zefri Idustri Egieerig Deprtet Shrif Uiversit of Techoog Tehr Ir (rzefri@hooco) Fuzz Sets Let X be cssic set of obects ced the uiverse whose geeric eeets re deoted b The ebership i crisp subset of X is ofte viewed s chrcteristic fuctio μ ( ) fro X to {0}such tht: μ ( ) = if = 0 otherwise where {0} is ced vutio set If the vutio set is owed to be the re iterv [0] is ced fuzz set proposed b Zdeh [] μ ( ) is the degree of ebership of i The coser the vue of μ ( ) is to the ore beog to Therefore is copete chrcterized b the set of ordered pirs: = {( μ ( )) X } The support of fuzz set is the crisp subset of X d is preseted s: Supp = { X μ ( ) > 0} α The α eve ( cut ) set of fuzz set is crisp subset of X d is deoted b α = { X μ ( ) α} fuzz set i X is cove if μ( λ + ( λ) ) i{ μ( ) μ( )} X d λ [0] tertive fuzz set is cove if α eve sets re cove ote tht i this pper we suppose tht X = fuzz uber is cove orized fuzz set o the re ie such tht ) It eists t est oe 0 with μ ( 0) = ) μ ( ) is piecewise cotiuous og the vrious tpes of fuzz ubers trigur d trpezoid fuzz ubers re of the ost iportt ote tht i this stud we o cosider trpezoid fuzz ubers fuzz uber is trpezoid fuzz uber if the ebership fuctio of it be i the foowig for: 8

2 Word cde of Sciece Egieerig d Techoog ) ( = + + ( β α) () where ( % = αβ ) F( ) L α L Fig Trpezoid Fuzz uber U U + β We show trpezoid fuzz uber b % = ( ) where the support of % is α β ( α + β ) d the od set of % is [ ] Let F( ) be the set of trpezoid fuzz ubers I the et subsectio we describe rithetic o F( ) rithetic o Fuzz ubers Let ( β = L U α ) d b = ( b L b U γ θ ) be two trpezoid fuzz ubers d The the resuts of ppig fuzz rithetic o the trpezoid fuzz ubers s show i the foowig: Ige of % : ( U L % = β α) dditio: L U + b = ( + b + b α + γ β + θ ) Scr Mutipictio: > 0 % = ( α β ) U L < 0 % = ( β α ) III KIG FUCTIOS coveiet ethod for coprig of the fuzz ubers is b use of rig fuctios rig fuctio is p fro F( ) ito the re ie ow we defie orders o F( ) s foowig: b if d o if ( % ) ( b % ) () > b if d o if ( % ) > ( b % ) () = b if d o if ( % ) = ( b % ) () where d b re i F ( ) It is obvious tht we write b if d o if b Sice there re rig fuctio for coprig fuzz ubers we o pp ier rig fuctios So it is obvious tht if we suppose tht be ier rig fuctio the i) b if d o if b 0 if d o if b ii) If b d c d the + c b + d Oe suggestio for ier rig fuctio s foowig: IV FUZZY LIE POGMMIG I this sectio we itroduce fuzz ier progrig (FLP) probes So we first defie ier progrig probes Lier Progrig ier progrig (LP) probe is defied s: M z = c st = b (5) 0 where c= ( c c ) ( ) T b= b b d = [ i ] I the bove probe of the preters re crisp [] ow if soe of the preters be fuzz ubers we obti fuzz ier progrig which is defied i the et subsectio Fuzz Lier Progrig Suppose tht i the ier progrig probe soe preters be fuzz ubers Hece it is possibe tht soe coefficiets of the probe i the obective fuctio techic coefficiets the right-hd side coefficiets or decisio ig vribes be fuzz uber [] [] [5] [6] [] [8] [9] Here we cosider the ier progrig probes with fuzz ubers i the obective fuctio V FUZZY UME LIE POGMMIG fuzz uber ier progrig (FLP) probe is defied s foows: M z = c st = b (6) 0 T where b c% ( F( )) d is ier rig fuctio Defiitio 5 We s tht vector is fesibe soutio to (6) if d o if stisfies the costrits of the probe Defiitio 5 fesibe soutio is opti soutio for (6) if for fesibe soutio for (6) we hve c c Fuzz sic Fesibe Soutio Here we itroduce bsic fesibe soutios for FLP probes Cosider the sste = b d 0 where is tri d b is vector ow suppose tht r( b) = r( ) = Prtitio fter possib rerrgig the cous of s [ ] where 85

3 Word cde of Sciece Egieerig d Techoog is osigur It is obvious tht r ( ) = The poit = ( ) where = b 0 = is ced bsic soutio of the sste If 0 the is ced bsic fesibe soutio (FS) of the sste Here is ced the bsic tri d is ced the obsic tri The copoets of re ced bsic vribes d the copoets of re ced obsic vribes If > 0 the is ced odegeerte bsic fesibe soutio d if t est oe copoet of is zero the is ced degeerte bsic fesibe soutio The foowig theore chrcterizes opti soutios The resut correspods to the so-ced odegeerte probes where fuzz bsic vribes correspodig to ever bsis re ozero (d hece positive) [5] Theore 5 ssue the FLP probe is odegeerte bsic fesibe soutio = b = 0 is opti to (6) if d o if z% c% for Proof Suppose tht = ( ) is bsic fesibe soutio to (6) where = b = 0 The z% = c% = c% b O the other hd for fesibe soutio we hve b= = + Hece we obti z% = c % = c% + c% = c% b ( c% ) i The z% = z% ( z% ) () i Therefore the resuts foow iedite fro () d the ssuptios of theore I the et sectio we propose sipe ethod for sovig FLP probes VI SIMPLEX METHOD FO THE FLP POLEMS Cosider the FLP probe s is defied i (6) z% = c% + c% st + = b 0 b + = Therefore Hece we write z ( c % + % c% ) = c% b Curret = 0 d the = b d z= c% b % The we rewrite the bove FLP probe i the foowig tbeu fort: z% z% 0 % HS The bove tbeu gives us the ifortio we eed to proceed with the sipe ethod The fuzz cost row i the bove tbeu isγ% = ( c % ) which cosists of the i % γ = z% c% s for the obsic vribes ccordig to the optiit coditio for these probes we re t the opti soutio if % γ 0 for i O the other hd if % γ < 0 for i the we echge with The we copute the vector the c be icrese idefiite d the the opti obective is ubouded O the other hd if hs t est oe positive copoet the the icrese i wi be boced b oe of the curret bsic vribes which drops to zero r = If 0 Theore 6 If i sipe tbeu eists such tht z% % < 0 d there eists bsic ide i such tht > 0 c 0 I % c% c the pivotig row r c be foud so tht pivotig o r wi ied fesibe tbeu with correspodig odecresig fuzz obective vue Proof We eed criterio for choosig bsic vribe to eve the bsis so tht the ew sipe tbeu wi rei fesibe d the ew obective vue is odecresig ssue cou is the pivot cou so suppose tht is = ( ) bsic fesibe soutio to the FLP probe where = b d = 0 The the correspodig fuzz obective vue is z% = c% b= c% O the other hd for bsic fesibe soutio to the FLP probe we hve + = 0 (8) i where = So if eters ito the bsis we write = 0 (9) Sice we wt be fesibe hece i 0 or i 0 for i = If i 0 the it is obvious tht the bove coditio is hod Hece for i > 0 we eed to hve % c b b i 0 86

4 Word cde of Sciece Egieerig d Techoog (0) i To stisf (0) it is sufficiet to et r0 = i{ i > 0} () r i so for bsic fesibe soutio to the FLP probe we hve z% = c% 0 ( z% ) () i So if we eter ito the bsis we hve z% = c% 0 ( z% ) () We ote tht the ew obective vue is odecresig sice z% = c% ( z% ) c% () 0 0 Usig the fct tht ( z ) 0 % Theore 6 If for bsic fesibe soutio to the FLP probe there is soe cou ot i bsis for which z% % < 0 d 0 i = the the FLP probe c i hs ubouded soutio Proof Suppose tht is bsic soutio to the FLP probe so + = i = = (5) or i i i = i 0 i i i i = = (6) ow if we eter ito the bsis the we hve > 0 d = 0 for i Sice i 0 i = hece i 0 () Therefore the curret bsic soutio wi rei fesibe ow the vue of ẑ for the bove fesibe soutio s foowig: So zˆ = c% + c% = c% ( ) + c% i i i = = c% ( c% ) i i i i = i = = c% 0 ( c% ) = z% ( z% c% ) zˆ = z% ( z% c% ) (8) Hece we c eter ito the bsis with rbitrri rge vue The fro (8) we hve ubouded soutio VII UMEICL EXMPLE For iustrtio of the bove ethod we sove FLP probe b use of sipe ethod Epe 8 M z% = (585) + (606) st We rewrite + + = = 0 0 We write the first fesibe sipe tbeu s foows: bsis HS z% (855) ( 0 66) 0 % 0 % 0 % Sice ( z% z% ) = (( 8 55 )( 0 66 )) d ( γ γ) = ( ( % γ) ( % γ)) = ( 5 8) the eters the bsis d the evig vribe is The ew tbeu is: bsis z% HS ( 6) 0 % 0 0 ( ) 0 % ( 0 ) ow fro( % γ % γ ) = (( 6)( )) d 5 ( ( % γ) (% γ)) = ( γ γ) = ( 6) it foows tht is eterig vribe d is evig vribe The st tbeu is show i the beow 8

5 Word cde of Sciece Egieerig d Techoog HS bsis z% 0 % 0 % ( ) ( ) ( ) w% = c% = ( c% c% ) = (( )( )) wb c 6 % = % b = ( ) ( c% b) = (% γ % γ ) = ( w % ) = (( )( )) 5 ( γ γ ) = ( ( % γ ) (% γ )) = ( ) > 0 % γ= % γ= 0 % ow usig optiit coditio there is ot vribe tht eters the bsis Therefore this bsis is opti VIII COCLUSIO We cosidered fuzz uber ier progrig probes d itroduced the bsic fesibe soutio for these probes Fi we obtied soe iportt resuts d we proposed ew gorith for sovig these probes direct b use of ier rig fuctio EFEECES [] MS zr JJ Jrvis d HD Sheri Lier Progrig d etwor Fows Joh Wie ew Yor Secod Editio 990 [] E e d L Zdeh Decisio ig i fuzz eviroet Mgeet Sci (90) --6 [] M Degdo JL Verdeg d M Vi geer ode for fuzz ier progrig Fuzz Sets d Sstes 9 (989) --9 [] SC Fg d CF Hu Lier progrig with fuzz coefficiets i costrit Coput Mth pp (999) 6--6 [5] Mhdvi-iri d SH sseri Duit i fuzz vribe ier progrig th Word Eforti Coferece WEC'05 Jue Istbu Ture [6] H Mei ig fuctios d their ppictios to fuzz ier progrig Fr Est J Mth Sci (00) 8--0 [] H Mei M Tt d M Mshichi Lier progrig with fuzz vribes Fuzz Sets d Sstes 09 (000) -- [8] H oefger Huschec d J Wof Lier progrig with fuzz obective Fuzz Sets d Sstes 9 (989) --8 [9] JL Verdeg du pproch to sove the fuzz ier progrig probe Fuzz Sets d Sstes (98) -- [0] H J Zier Fuzz progrig d ier progrig with sever obective fuctios Fuzz Sets d Sstes (98)