Fuzzy Goal Programming for Solving Fuzzy Regression Equations

Transcription

1 Proeedngs of he h WSEAS Inernaonal Conferene on SYSEMS Voulagmen Ahens Greee July () Fuzzy Goal Programmng for Solvng Fuzzy Regresson Equaons RueyChyn saur Dearmen of Fnane Hsuan Chuang Unversy 8 Hsuan Chuang Road Hsnhu awan HsaoFan Wang Dearmen of Indusry Engneerng and Engneerng Managemen aonal sng Hua Unversy Hsnhu awan. Absra he range of a fuzzy regresson nerval s deded by he olleed daa and he onfdene level h. I s undersood ha a larger value h n a fuzzy regresson equaon mles a larger fuzzy regresson nerval. In order o examne he effe of value h Moskowz and Km develoed an analyal mehod o assess he shae and range of he ossbly dsrbuon of a membersh funon n order o reveal more relable and reals resuls from he fuzzy regresson. If he ossbly of eah daum n he fuzzy regresson nerval an be easly found hen he regresson analyss should be arred ou resely. However s omlaed o fnd a roer value h among he yes of membersh funons of fuzzy arameers and olleed fuzzy daa as roosed by he analyss roess of Moskowz and Km. herefore n hs sudy a fuzzy goal rogrammng mehod s roosed for solvng a fuzzy regresson equaon n whh a maxmum sasfaory value h n he fuzzy regresson nerval an be solved. umeral examles are rovded o llusrae our roosed mehod. Keywords: Fuzzy regresson foreasng lnear rogrammng fuzzy goal rogrammng value h.. IRODUCIO Fuzzy regresson models have been aled o varous areas suh as model exenson [7][][] [8] busness foreasng [][] and engneerng [][]. When esablshng a fuzzy regresson model seleng a roer value of onfdene level h s moran beause deermnes he range of he ossbly dsrbuon of he reded fuzzyouu value. As a resul hs roblem has drawn muh aenon from researher anaka and Waada [] suggesed ha when he olleed daa s suffenly large h an be used and a larger value h should be adoed f he sze of he olleed daa s small. Bardossy e al [] hen suggesed o sele a value h aordng o he deson maker s belef n he model and reommended he value h o be beween. and.7. However hese suggesons are ofen vague and raher dfful o usfy or aly n real lfe suaon In order o oe wh hs roblem Moskowz and Km [8] develoed an analyal mehod o assess he shae and range of he ossbly dsrbuon of a membersh funon n order o show more relable and reals resuls from he fuzzy regresson nerval. If he ossbly of eah daum n he fuzzy regresson nerval an be easly found hen regresson analyss should be able o be arred ou n a more resely fashon. However s que omlaed o analyze he membersh funons n order o fnd a roer value h. herefore n hs sudy we roose a fuzzy goal rogrammng mehod n order o solve a fuzzy regresson equaon n whh a maxmum sasfaory value h n he fuzzy regresson nerval an be solved. In Seon we wll dsuss how anaka s model s affeed by boh value h and he olleed daa. In Seon fuzzy goal rogrammng s nrodued and n Seon a fuzzy regresson model s derved from a nonreemve FGP. In Seon he examles for dfferen suaons of olerane levels are gven. Fnally we draw our onluson n Seon.. FUZZY REGRESSIO MODEL anaka e al [] was he frs o sudy he ssue of a fuzzy regresson model wh rsnu and fuzzyouu value he fuzzy regresson model s an alernave aroah o evaluang he fuzzy rela

2 Proeedngs of he h WSEAS Inernaonal Conferene on SYSEMS Voulagmen Ahens Greee July () on beween ndeenden varables and deenden varable Km e al [7] also showed ha s foreasng error s beer han ha of sasal regresson f he olleed daa are smaller by smulaon and omarson wh hose wo model Insead of usng on esmaon as n he onvenonal robably heory he fuzzy regresson model s used o granulae a one no a se wh membersh funon hereby dereasng he amoun of requred daa. he bas model s assumed o be a fuzzy lnear funon as follow Y A A... A... A A () where [.. ] s a veor of ndeenden varables; A [ A A... A... A ] s a veor of he fuzzy arameer resened n he form of a symmer rangular membersh funon. Le us assume a fuzzy arameer A ( α ) wh s enral value α and s sread value hen s membersh funon s defned as () below. α a u ( ) a α a α A () oherwse In addon eah olleed fuzzyouu daum. M s assumed o be a fuzzy number denoed as Y ( y e ) where y s a ener value and e s s sread value and for smly s membersh funon s defned as a symmer rangular shae. y y y e y y e u ( y ) e () Y o. w. herefore model () an be rewren as Y ( α ) ( α )... ( α ) () By alyng he Exenson Prnle [] he derved membersh funon of fuzzy number Y s shown n () and eah value of he deenden varable an be esmaed as a fuzzy number L h U Y ( Y Y Y ). M where he lower bound of Y s Y ( α ) L ; he ener value of Y h s Y α and he uer bound of Y U s Y ( α ) where [... ] α α α.... [ ] α y α u ( y ) y Y () o. w. he degree of fness of he esmaed fuzzy lnear model Y A Y y e s o he gven daa ( ) h Y measured by whh s he maxmum value of h h h sube o Y Y where h Y { y u ( y ) h Y } h and Y { y u ( y ) h}. he ndex s loed n Fgure. hen he vagueness of he fuzzy lnear regresson s obaned as M J () ex by nerval arhme he membersh funon of obaned from Fgure s derved as h h y u h α. M (7) Y e e y k α Fgure Degree of fness of Y h y Y o olleed daa Y If he oal vagueness J s mnmzed and sube o fndng a maxmum membersh degree h where eah olleed daa s nluded n he fuzzy regresson nerval wh a leas a membersh degree h as h h hen he above analyss leads o he followng lnear rogrammng model. M Mn α α x ( h) ( h) α R h ; y ( h) e y ( h) e... M... From (8) s evden ha he nerval of fuzzy (8)

3 Proeedngs of he h WSEAS Inernaonal Conferene on SYSEMS Voulagmen Ahens Greee July () regresson whh s derved from anaka s mehod s deermned by he olleed daa and he value h. Beause he larger y value or h value wll resul n a larger fuzzy regresson nerval f he value h s msused he nerval s usually wde and unredable.. FUZZY GOAL PROGRAMMIG Goal rogrammng (GP) s an a deson n modelng real world deson roblems and has been exensvely used n solvng deson makng roblem However a maor lmaon of GP for a deson maker s he mreson of he goal level asred o. hus n fuzzy goal rogrammng (FGP) fuzzy goals are onsdered wh mrese levels [][][9][]. FGP was frs nrodued by arasmhan [9] who solved a FGP roblem by assumng of lnear membersh funons nvolvng o solve a se of k lnear rogrammng roblems wh equal and unequal fuzzy weghs and eah goal onanng k onsrans wh k denong he number of goals n he orgnal roblem. Aferwards Hannan [] smlfed he roedure o formulae a sngle LP roblem wh k goalrelaed onsrans whh an be reemve or nonreemve. herefore when fuzzy goals are resened as essenally equal o b a FGP roblem an be wren n a general form as Ax b Cx d (9) x where x s an n alernave se A s a k n marx of oeffens of obeve funon C s a k n marx of oeffens of onsrans and d s rghhand sde wh a k marx. he membersh funon of he fuzzy relaon n fuzzy goals an be formulaed n a rangular form as () ( A) b ( ) [( b ) ( A) ] u A b ( A) b () [ ( A) ( b )] b ( A) b ow. When he fuzzy deson s made by sasfyng all of he goals o he larges degree he maxmn oeraor s used suh ha he omal deson D s obaned by u x Max Mn u A () D ( ) ( ) x For a FGP roblem Hannan [][] roved ha f λ s he omal soluon o he subroblem wh u ( A ) λ hen here exss λ suh... ha he omal soluon n (9) wll equal o λ. herefore model (9) an be wren as (). ( A ) Cx b λ d d [ ] d d d d e... K... K ;... K k () However he membersh funon of Hannan s model s omuaonally omlaed and arasmhan s model anno deermne he rores of he fuzzy goal herefore war e al. [] demonsraed an algorhm for solvng a FGP roblem wh symmeral rangular membersh funons of fuzzy goals and rory sruure. Aferwards Chen [] Wang and Fu [7] onnued o nvesgae more effen mehods from he roeres of hese FGP model. FORMULAIO OF A FUZZY REGRESSIO BY FGP When analyzng a fuzzy regresson model s undersood ha s obeve s o mnmze he oal vagueness of he derved fuzzy regresson as M mn n whh eah daa has s or resondng level of fuzzness. If a deson maker (DM) desres o oban a mnmum fuzzy regresson nerval hen he degree of fuzzness e of every olleed ouu daa Y ( y e ) should be as lose o as ossble. hen desred o be aroxmaely equal o e hus we an onsder e...m s as he mrese goal Le denoe he olerane nerval for goal e. he fuzzy goal an be desrbed by a rangular membersh funon as (). In order o oban he maxmum membersh degree for he goal model () s requred o be greaer han membersh level λ suh ha u ( ) λ.

4 Proeedngs of he h WSEAS Inernaonal Conferene on SYSEMS Voulagmen Ahens Greee July () e ( e ) u ( ) e e () ( e ) e e ow. Besdes from he on of membersh level a DM should smulaneously evaluae he membersh degree of every olleed daa n fuzzy regresson nerval o be greaer han λ ha s h λ. hen model (8) s rewren as model () below Max Mn λ λ ( ) α α λ [ ] ( λ ) ( λ ) ( λ ) α R... M... e... M y y e e ; () Furhermore by nrodung a slak varable as a negave devaon varable and leng ( λ ) d be he osve devaon varable n he frs onsran of model () model () an be rewren no () as a form of fuzzy goal rogram. Max Mn λ λ α ( ) α λ d [ ] d ( λ ) d ( λ ) e d y e α R... M M y e However beause mn( λ λ ) ( ) λ λ.hs mles ha d () λ herefore d λ d hus () an be ransformed no model (). α α λ d [ ] d d d ( λ) ( λ)... M d e y e α R... M... From onsrans and n model () we y e have ( ) e ; () λ M herefore he value of d n model () s always zero. By subsung onsran no onsrans and of model () we have model (7). α α λ d [ ] λe ( λ) d λe ( λ) d d... M α R y y... M... Sne afer solvng (7) for values α we need o solve ; (7)... we sugges usng model (8) below o smulaneously solve boh varable α α λ d [ ] d λe ( λ) d λe ( λ) d... M d e y y α R... M... Model (8) s a nonlnear roblem whh an be solved by GIO. Besdes he meanng of value λ n model (8) s smlar o he value h n model (8). Model (7) mles ha he larger he requred membersh degree h he wder he regresson nerval ; (8)

5 Proeedngs of he h WSEAS Inernaonal Conferene on SYSEMS Voulagmen Ahens Greee July () and also ha he larger he value of λ n () he larger he fuzzy regresson nerval. ha s f a DM s oms o he olleed daa hen a smaller value wll be onsdered o derve a smaller value λ whh makes a smaller regresson nerval. However value h n (8) s deded by a DM whh normally s no easy o esmae roerly however λ s deermned by he model from mlly radeoff he weghs of all olleed daa. hus he roosed model s useful when value h s unknown by he DM. hus he derved fuzzy regresson s Y (..8) (.. 7). o omare he derved fuzzy regresson wh anaka s model a value h. as suggesed by Bardossy Bogard and Duksen [] s onsdered. Fgure shows he resuls of he roosed mehod beng able o derve a regresson nerval whh no only onans all olleed daa when DM does no know he value of h; bu also s narrower han ha of anaka s fuzzy regresson model. he Proosed model n (8). anaka s model olleed daa.. A ILLUSRAED EAMPLE An examle adoed from P. Damond [] s shown n able. able. Colleed Daa [] Y y e ( ) 7 8 (..8) (..) (..) (..) (..) (..7) (..8) (..) In hs exermen we show how o derve a fuzzy regresson nerval by FGP when he onfdene level h s no rovded. In arular we wll demonsrae how he olerane value affes he regresson resul Frs we onsder a se of daa n able where dfferen olerane levels are gven for dfferen daa. By alyng model (8) he soluon s obaned by α. α..8.7 where...9 d d d d wh membersh level of λ.7. able Values d d d d Fgure. Comarson of anaka model and he FGP mehod Seond by hangng dfferen olerane levels o be unfed levels suh as. M we have he soluon of α.8 α..78 where d d d d d wh a membersh level of λ.7. Fgure ombnes hese wo suaons for omarson and shows ha a unfed level of wll lead o a larger regresson nerval whh resuls from he fa ha he DM s unable o dfferenae eah daum n he fuzzy goal As a resul a hgher degree of unerany s embedded n he fuzzy regresson nerval Fgure. Comarson of ondon and ondon n hs ase. Conluson In hs sudy we roosed a fuzzy goal rogrammng model o solve a fuzzy regresson equaon when he requred membersh degree h of eah d d 8 d he frs ondon n (8). he seond ondon n (8). olleed daa.

6 Proeedngs of he h WSEAS Inernaonal Conferene on SYSEMS Voulagmen Ahens Greee July () daum s no known. he roosed models need o deermne he olerane value a ror whh s somemes dfful by a DM. If hs s he ase equal s suggesed whh resuls n a wder regresson nerval. Furher nvesgaon of hs model for when here exss an ouler n he olleed daa an furher nrease he redon ably of he model. Aknowledgemen he auhors aknowledge he fnanal suor from aonal Sene Counl wh roe number # SC 88E7. Referenes [] A. Bardossy I. Bogard and L. Duksen Fuzzy regresson n hydrology Waer Resoures Re Vol. o [] P.. Chang S. A. Konz and E. S. Lee Alyng fuzzy lnear regresson o VD legbly Fuzzy Ses and Sysems Vol.8 o [] H.K. Chen A noe on a fuzzy goal rogrammng algorhm by war Dharmar and Rao Fuzzy Ses and sysems Vol. o [] P. Damond Fuzzy leas square Informaon Senes Vol. o [] E.L. Hannan On fuzzy goal rogrammng Deson Senes Vol. o [] E.L. Hannan Conrasng fuzzy goal rogrammng and fuzzy mulrera rogrammng Deson Sene Vol. o [7] K.J. Km H. Moskowz and M. Koksalan Fuzzy versus sasal lnear regresson Euroean Journal of Oeraonal Researh Vol.9 o [8] H. Moskowz k. Km On assessng he H value n fuzzy lnear regresson Fuzzy Ses and Sysems Vol.8 o [9] R. arasmhan Goal rogrammng n a fuzzy envronmen Deson Senes Vol [] G. Peers Fuzzy lnear regresson wh fuzzy nervals Fuzzy Ses and Sysems Vol. o [] H. anaka and S. Uema and K. Asa Lnear regresson analyss wh fuzzy model. IEEE ran S.M.C. Vol. o [] H. anaka Fuzzy Daa Analyss by Possbly Lnear Models Fuzzy Ses and Sysems Vol. o [] R.. war S. Dharmar and J.R. Rao Prory sruure n fuzzy goal rogrammng Fuzzy Ses and Sysems Vol.9 o [] R.C. saur H.F. Wang Oulers n Fuzzy Regresson Analyss Inernaonal Journal of Fuzzy Sysems Vol. o. (999) 9. [] R.C. saur H.F. Wang J.C. O Yang Fuzzy Regresson for Seasonal me Seres Analyss Inernaonal Journal of Informaon ehnology & Deson Makng Vol. o..7. [] R.C. saur Exraolae Inerne Users n awan by Rsk Assessmen Comuers and Mahemas wh Alaons Vol. o. () 77. [7] H.F. Wang C.C. Fu A generalzaon of fuzzy goal rogrammng wh reemve sruure Comuers and Oeraons Researh Vol. o [8] H.F. Wang R.C. saur Insgh of a Fuzzy Regresson Model Fuzzy Ses and Sysems Vol. o..9. [9] H.F. Wang R.C. saur BCrera Varable Seleon n Fuzzy Regresson Equaon Comuers and Mahemas wh Alaons Vol. o [] H.F. Wang R.C. saur Resoluon of fuzzy regresson model Euroean Journal of Oeraon Researh Vol. o..7. [] L.A. Zadeh Fuzzy ses Inform. Conrol Vol.8 o [] H.J. Zmmermann Fuzzy Ses Deson makng and Exer Sysems Kluwer Aadem Boson 987. [] H.J. Zmmermann Fuzzy rogrammng and lnear rogrammng wh several obeve funon Fuzzy Ses and Sysems Vol. o