MODUL KULIAH SUDRADJAT

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1 MODUL KULIAH SUDRADJAT JURUSAN MATEMATIKA FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM UNIVERSITAS PADJADJARAN BANDUNG 8

2 KATAPENGANTAR Modul kulh dsusu sebg pelegkp buku text kulh tetg Logk Fuzzy, yg dberk utuk mhssw progrm stud Mtemtk d tgkt sr. Meggt mter Logk Fuzzy memerluk pegthu dsr mege hmpu fuzzy mk modul kulh dsusu deg urut pertm pemhm tetg kosep hmpu fuzzy, kemud pemhm tetg logk fuzzy d terkhr peggu hmpu fuzzy pd pemogrm ler d seklgus pemhm tetg pemodel. Pd bg wl, k dbhs tetg hmpu crsp, hmpu fuzzy yg merupk dsr-dsr dr opers logk fuzzy. Cotoh-cotoh hmpu crsp, fugs keggot d kosep possblstk. Bg kedu, k dbhs Fuzzy logc, serh perkembg fuzzy logc, hmpu crsp d fuzzy d Vlds d kosstes pd fuzzy logc. Bg khr, k dbhs fuzzy pemogrm ler, Iterctve fuzzy pemogrm ler, Algortm Iterctve pemogrm ler fuzzy d dsr-dsr pemodel mtemtk. Modul dsusu utuk pertm kl, mudh-mudh modul dpt memberk rh dlm mempelr logk fuzzy khususy bg pr mhssw d dhrpk d msuk-msuk utuk perbk sehgg pd khry modul bs dterbt dlm betuk buku. Bdug, Agustus 8 Peuls

3 DAFTAR ISI KATA PENGANTAR DAFTAR ISI BAB I PENDAHULUAN BAB II HIMPUNAN CRISP 3. Pedhulu 3. Propert dr opers hmpu crsp Kosep dsr d termology hmpu fuzzy... 4 BAB III HIMPUNAN FUZZY Pegert hmpu fuzzy Fugs keggot Teor possblstk Trpezodl blg fuzzy 8 BAB IV FUZZY LOGIC 3 4. Fuzzy logc 3 4. Serh perkembg fuzzy logc Crsp Set d Fuzzy Vlds d kosstes pd fuzzy logc 9 BAB V PEMOGRAMAN LINIER FUZZY Fuzzy pemogrm ler Iterctve fuzzy pemogrm ler Algortm Iterctve pemogrm ler fuzzy BAB VI DASAR-DASAR PEMODELAN Kosep dsr sstem Sft dsr system Perkembg kesstem Pemodel Mtemtk Keutug dr pemodel Klsfks model Klsfks model ltk Krkterstk model yg bk Proses pegem model DAFTAR PUSTAKA

4 BAB I PENDAHULUAN Hmpu fuzzy mempuy per yg petg dlm perkembg mtemtk khususy dlm mtemtk hmpu. Mtemtkw Germ George Ctor (845-98) dlh org yg pertm kl secr forml mempelr kosep tetg hmpu, Jtze [7]. Teor hmpu sellu dpelr d d terpk sepg ms, bhk smp st mtemtkw sellu megembgk tetg bhs mtemtk (teor hmpu). Byk peelt-peelt yg megguk teor hmpu fuzzy d st byk lterturelteltur tetg hmpu fuzzy, msly yg berkt deg tekk cotrol, fuzzy logc d rels fuzzy. Ide hmpu fuzzy (fuzzy set) d wl dr mtemtk d teor system dr L.A Zdeh [35 ], pd thu 965. k dteremhk, fuzzy rty tdk els/burm, tdk pst. Hmpu fuzzy dlh cbg dr mtemtk yg tertu, yg mempelr proses blg rdom: teor problts, sttstk mtemtk, teor forms d ly. Peyeles mslh deg hmpu fuzzy lebh mudh dr pd deg meguk teor probblts (kosep pegukur). Fuzzy Logc dpt dktk sebg logk bru yg lm, sebb lmu tetg logk moder d metods bru dtemuk pd thu 965, pdhl sebery kosep tetg fuzzy logc tu sedr sudh d sek lm. Slh stu cotoh peggu fuzzy logc pd proses put-output dlm betuk grfs sepert pd Gmr., Kusumdew []. Beberp ls dguky fuzzy log : (Kusumdew [], Sudrdt [9] Y, Ry d Power [34]), dlh. Kosep fuzzy logc mudh dmegert.. Fuzzy logc sgt fleksbel. 3. Fuzzy logc memlk toles terhdp dt yg kurg tept, Popescu, Surdt d Ghc [5, 6]

5 4. Fuzzy logc mmpu memodelk fugs oler yg kompleks. 5. Fuzzy logc ddsr pd hs lm. Fuzzy Logc st byk dterpk dlm berbg bdg, Jtze [7], dtry: Fuzzy rule Bsed Systems Fuzzy Noler Smultos Fuzzy Decso Mkg Fuzzy Clssfcto Fuzzy Ptter ecogto Fuzzy Cotrol Systems Sebg cotoh perhtk proses put-output sepert pd gmbr. INPUT OUTPUT Persed brg Akhr mggu KOTAK HITAM Gmbr. Poses put out-put Modul terdr dr 6 bb, ytu Bb Pedhulu, Bb, Hmpu crsp yg terds dr kosep dsr d termolog hmpu fuzzy, Bb 3, Pegert fuzzy, fugs keggot, teor possblstk, trpezodl blg fuzzy, Bb 4 membhs tetg fuzz logc, serh perkembg fuzzy logc, vlds d kosstes pd fuzzy logc, Bb 5 membhs tetg, pemogrm ler fuzzy, terktf pemogrm ler fuzzy, lgortm terktf pemogrm ler fuzzy, d Bb 6 membhs ttg dsr-dsr pemodel d klsfks model.

6 BAB II HIMPUNAN CRISP. Pedhulu Jk x dlh ggot tu eleme dr hmpu A, kt tuls x A, d k x dlh buk ggot tu eleme dr hmpu A, kt tuls x A. Hmpu A deg ggot,..., dotsk A = {,..., A }, hmpu B yg memeuh property P,..., P dtuls B = { b b propertes P,..., P }, dm symbol meotsk sedemk sehgg. Petg d serg dguk pd vector spce Euclde R, rel umbers. Hmpu A pd R dktk covex k, utuk setp ttk r = r N ) d s = s N ) pd A ( ( d setp blg rel tr d, exclusve, ttk t = r + ( ) s N ) ug ( dlm A, deg kt l hmpu A pd R dlh covex k, utuk setp ttk r d s pd A, semu ttk terletk pd segme grs terkoeks r d s ug pd A.. Propert dr opers hmpu crsp Ivoluto Comuttvty Assoctvty Dstrbutvty Idempotece A = A A B) = B A A B = B A ( A B) C) = A ( B C) A ( B C) = A ( B C) A ( B C) = ( A B) ( A C) A ( B C) = ( A B) ( A C) A A = A A A = A

7 Absorpto Absorpto of compoet Absorpto by X d φ Idetty Lw of cotrdcto Lw of excluded mddle DeMorg s lws A ( A B) = A A ( A B) = A A ( A B) = A B A ( A B) = A B A X = X A φ = φ A φ = A A X = A A A = φ A A = X A B = A B A B = A B.3 Kosep dsr d termology hmpu fuzzy Pd bg k dkemukk tetg kosep dsr d termology dr hmpu fuzzy. Eleme-eleme dr hmpu fuzzy dmbl dr hmpu uversl dr sstem yt secr lus tu secr terbts. Uversl memut semu eleme, sebg cotoh, Jtze [7]. Hmpu uversl hádl mus yg tergolog us mud yg berumlh tr d, d d refresetsk pd gmbr. b. Hmpu x dktk uversl dr semu pegukur postf. Gmbr. Pegelompok us yg dbg berdsrk mud d tu Byk pegembg d geerlss dr kosep dsr dr hmpu crsp. Sebg lustrs Klr d Folger [9] dr beberp kosep, kt berk dert membershp dr eleme-eleme

8 hmpu uversl kedlm empt hmpu fuzzy yg berbed sepert terlht pd tble. d secr grfk terlht pd Grfk.. Hmpu uversl X dr umur yg dkelompok sebg berkut: X = {5,,, 3, 4, 5, 6, 7, 8} d fuzzy set d defsk berdsrk k-k, dews, mud, d tu dlh 4 eleme dr hmpu kus (power set) dr semu possble fuzzy subsets dr X. Support dr hmpu fuzzy A pd hmpu uversl X dlh hmpu crsp yg memut semu eleme-eleme dr X d dert membershp pd A tdk ol. Support hmpu fuzzy pd X dytk deg fugs Supp : F( X ) F( X ), (.) dm Supp A = { x X μ ( x) > }. (.) A Dr Tbel., support fuzzy set Mud dlh crsp set Supp(Mud) = {5,,, 3, 4, 5} Nots khusus yg kdg-kdg dguk utuk medefsk hmpu fuzzy deg fte support. Asumsk bhw x dlh eleme dr support hmpu fuzzy A d μ dlh dert membershp pd A. Mk A dpt dtuls: μ μ μ A = + + L+. (.3) x x x Tbel. Eleme-eleme uversl Eleme Akk Dews Mud Tu (umur)

9 ,,8,6,4, Ak-k Dews Mud Tu Eleme (umur) Gfk. Fugs keggot Utuk ksus dm hmpu fuzzy A ddefsk dlm hmpu uversl yg terbts d terhtug, dpt d tuls: μ A = = (.4) x Sm hly, k X pd tervl dr blg rl, hmpu fuzzy A dtuls dlm betuk: A = μ ( x) / x. (.5) x A Defs. Klr d Folger [9] Heght of fuzzy set dlh eleme-eleme dr sutu hmpu fuzzy yg mecp dert membershp terbesr. Defs. Klr d Folger [9] Hmpu fuzzy dsebut dormlss (ormlzed) dm eleme-eleme merupk kemugk mksmum dr dert keggot. Jk rge dert membershp merupk tervl tertutup tr d, mk slh stu eleme dr dert membershp yg termut pd ormlss, sedgk heght dr hmpu fuzzy dlh. Sebg cotoh perhtk Tble. tg hmpu fuzzy dews,

10 mud d tu, Gmbr. d.3, dlh semuy dormlss, d heght dlh sm deg.

11 Hmpu fuzzy A dlh orml k Heght ( A) = Mxx A( x) =, sepert terlht pd bmbr berkut: Gmbr.4 Betuk trpezodl Defs.3 Klr d Folger[9] α -cuts dr hmpu fuzzy A dlh hmpu crsp A α yg memut semu eleme dr hmpu uversl X yg mempuy dert membershp pd A leh besr tu sm terhdp l α. Ddeefsk: A = { x X μ A ( x) α}. (.6) α Defs.4 Klr d Folger [9] Utuk μ F(X ) d α [,]. Mk hmpu [ ] = { x X μ( x) α} dsebut α -cut tu hmpuα -level dr μ. μ α Sebg lustrs, perhtk Tbel.., utuk α =. Mud. ={5,,, 3, 4}

12 Deg cr yg sm bs d cr utukα =. 8 d α =. Cotoh dr hmpu fuzzy seprt pd gmbr. Berk μ F ( X ), α [,], β (,). Teorem. Negoţă [4] Berk μ F ( X ), α [,], β (,).. [μ = X ] b. α < β [ μ] α [ μ] β c. I[ μ] α = [ μ] β utuk semu β [,]. α: α < β Teorem. Negoţă [4] Ambl F(X ) μ, mk μ( t) = sup { α I ( t) } α = [,] μ α utuk semu μ F(X ). Defs.4 Klrr d Folger [9] X dlh rug vektor. Hmpu fuzzy μ(x ) dlh fuzzy koveks k semu α cuts dlh hmpu koveks. Defs.5 Negoţă [4] Sutu hmpu fuzzy dlh covex k d hy k setp α - cuts dlh hmpu covex. Sutu fuzzy set A dlh covex k d hy k μ ( r + ( r) s) m[ μ ( r), μ ( s)], r, s R d [,] (.7) A Teorem.3 Negoţă [4] μ dlh fuzzy koveks x, x X [,] A A : μ( x + ( ) x ) μ( x ) μ( x ). (.8) Defs.6 Klr d Folger [9] Sklr crdlty dr hmpu fuzzy A pd hmpu uversl terbts X dlh umlh dr dert keggot dr semu eleme X d A, dtuls

13 A = μ ( x). (.9) x X A Sclr crdlty pd hmpu fuzzy s Tu dr Tble. d ts dlh Tu = = 4. Sclr crdlty pd fuzzy set k-k dlh. Betuk l dr crdlty dlh fuzzy crdlty A dlh fuzzy set (fuzzy umber) ddefsk dlm N dm fugs keggot dlh μ (Aα ) = α. (.) A Utuk semu α dlm level set dr A. T ~ u = = Defs.7 Klrr d Folger [9] Jk dert membershp pd setp eleme dr hmpu uversl X pd fuzzy set A dlh lebh kecl tu sm deg dert membershp pd fuzzy set B, mk A dsebut subset dr B k d hy k μ ( x) μ ( x), utuk setp x X, mk A B. Hmpu fuzzy tu pd Tbel. dlh hmpu bg dr dews kre utuk setp eleme d d dlm hmpu uversl μ ( x) μ ( x). (.) tu dews Defs.8 Klr d Folger [9] Hmpu fuzzy A dktk sm deg hmpu fuzzy B μ ( x) = μ ( x), x X, d dtuls A = B. A B Jelsy, k A = B, mk A B d A B. Jk μ ( x) μ ( x), x X A A B B, d dtuls A B. Defs.9 Klr d Folger [9] Hmpu fuzzy A dktk proper subset dr hmpxu fuzzy B dm A dlh subset dr B d du hmpu dlh tdk sm, mk μ ( x) μ ( x), x X, d μ ( x) < μ ( x), x X d dotsk A B k d A hy k B A B d A B. A B

14 Defs. Klr d Folger [9] Rge dert keggot dlm tervl tertutup tr d, dsebut compleme dr hmpu fuzzy yg bersesu deg hmpu uversl X d otsk A d ddefsk μ ( x) = μ ( x), X. A Hmpu fuzzy tdk tu dr Tbel. dlh tdk tu = 5 A Defs. Klr d Folger [9] A d B dlh du hmpu bg dr hmpu fuzzy A B. mud tu = Defs. Klr d Folger [9] Irs dr du hmpu fuzzy A d B dlh hmpu A B sedemk hgg μ A B( x) = m[ μ A( x), μb ( x)], X. (.).... mud tu = Defs.3 Klr d Folger [9] Jk fugs f memetk ttk-ttk pd hmpu X pd ttkttk pd hmpu Y d sutu hmpu fuzzy A P(X ), dm μ μ μ A = + + L + x x x exsteso prcple sttes :, (.) f ( A) = = μ μ μ f + + L + x x x μ μ μ + + L + f ( x ) f ( x ) f ( x. ) (.3) Defs.4 Hmpu fuzzy A dlh koleks psg berurut A = {( x, μ( x))}

15 Item x dlm uversl d μ (x) dert keggot dr A. Psg berurut tuggl ( x, μ ( x)) dsebut sgleto fuzzy; dlm betuk vektor = μ( x ), μ( x ), L, μ( x )) (.4). ( Perlu dperhtk bhw setp poss (,, L, ) berkorespodes terhdp ttk-ttk pd uversl.

16 BAB III HIMPUNAN FUZZY 3. Pegert hmpu fuzzy Hmpu fuzzy pertm kl dkembgk pd thu 965 oleh Zdeh [47], teor hmpu fuzzy telh byk dkembgk d d plksk dlm berbg mslh rel. Kosep hmpu fuzzy yg dkembgk oleh Zdeh [35] Defţ 3. Bodg [3] Perhtk X dlh hmpu uversl. Mk hmpu bg fuzzy A dr X ddefsk deg fugs keggot (membershp fucto) μ A : X [,] (3.) dm setp eleme x X d blg rel μ (x A ) pd tervl [,], dm l μ (x A ) meuuk tgkt keggot (membershp) dr x pd A. Hmpu fuzzy dr A ddefsk: A = {( x, μ A( x)) x X} (3.) Defs dpt dgeerlssk k tervl tertutup [,] dlh dgt deg eleme mksmum tu mmum. Perhtk A, B X du hmpu fuzzy deg fugs keggoty μ (x A ) d μ (x) B. Ktk bhw A dlh hmpu bg dr B, otsk A B, k d hy k μ A( x) μb( x), x X (3.3) Dr defs dperoleh bhw A dlh sm deg B, dotsk A = B, k d hy k μ ( x) = μ ( x), x X (3.4) A B Kompleme A dr hmpu fuzzy fuzzy A ddefsk μ ( x) = μ ( x), x X (3.5) A A

17 Gbug du hmpu fuzzy A d B dlh hmpu fuzzy deg fugs keggoty μ ( x) = mx( μ ( x), μ ( x)) = μ ( x) μ ( x), x X (3.6) A B A D fugs keggot dr rs du hmpu fuzzy A d B dlh B A μ ( x) = m( μ ( x), μ ( x) = μ ( x) μ ( x), x X (3.7) A B A B A B B Gmbr 3. : Irs d Gbug du hmpu fuzzy Defs 3. Bodg Lu[3] Hmpu eleme-eleme dr hmpu fuzzy A yg plg kecl dr tgkt keggot α, dsebut α -level set, dotsk A = { x X μ A ( x) α}. α Secr khusus, kt sebut fuzzy umber(fuzzy qutty) sutu fuzzy subset ~ dr rl r deg fugs keggot μ ~ : r [,]. Ambl ~ d b ~ du blg fuzzy deg fugs keggot berturut-turut μ ~ d μ b ~.

18 3.. Fugs keggot Terdpt du defs fugs keggot (membershp fucto) utuk hmpu fuzzy: Numercl d fuctol. umercl defsk peryt tgkt dr fugs keggot dr hmpu fuzzy dytk deg vector blg to expresses the degree of membershp fucto of fuzzy set s vector of umbers whose dmeso depeds o the level of dscretzto.,.e the umber of dscrete elemets the uverse. Fuctol ddefsk deg meetuk fugs keggot dr hmpu fuzzy dlm peryt ltk yg meytk tgkt keggot utuk setp eleme yg dtetuk pd hmpu uversl of dscourse to be clculted. Stdr tu shpes dr fugs keggot dlh kesepkt yg dguk utuk dsr hmpu fuzzy pd uversl U dr blg rl.fugs keggot yg serg dguk dlh :() S-fucto, (b) π -fucto, (c) trgulr form (d) trpezod form d (e) expoetl form, Klr d Folger [9] Fugs S: for [( u ) /( c )] for S( u; ; b; c) = [( u c) /( c )] for for u < u b b u c u > c Fugs π S( u, c b, c b /, c) π ( u; b; c) = S( u; c, c + b /, c + b) for for u c c c

19 Fugs segtg ( u ) /( b ) T ( u; ; b; c) = ( c u) /( c b) for for for for u < u b b u c u > c 3.3. Teor possblstk Fugs keggot fuzyy dlh berbed deg dstrbus probblts sttstk. Sebg lustrs berkut yg dsebut egg-etg exmple, Jtze [7], Tk, Guo d Türkse [3] (Zdeh Zmmerm [35] ) Berkut peryt Hs mk X telor utuk srp pg, dm X U = {,,3,4,5,6,7,8 }. Ak dperlhtk soss dstrbus probblts p deg observs Hs mk srp pg utuk hr, U = [ ] p = [..8. Hmpu fuzzy megekspresk dert dr ksus deg peryt bhw Hs dpt mk X telor dsebut dstrbus possblstk π : U = [ ] p = [ Dm possblstk utuk X = 3 dlh, d probllts dlh hy. Dsr dr kosep d tekk dr teor possblty dkemukk oleh Zdeh [36], possblty dr lebh kecl tu sm deg b ddefsk sebg berkut: Dubos d Prde [3], ~ Pos ( ~ b ) = sup{m( μ ~ ( x), μ~ ( y)) x, y r, x y}, (3.9) b dm Pos dlh posblty. Deg kt l bhw posblstk ~ ~ b dlh lebh besr dm terdpt lebh kecl dr l x, y r sedemk sehgg x y, d l dr ~ d b ~ berkorespodes deg x d y. Deg cr yg sm utuk, posblstk ~ ~ < b ddefsk ].]

20 ~ Pos ( ~ < b ) = sup{m( μ ~ ( x), μ~ ( y)) x, y r, x < y} ), (3.) b Posblstk ~ ~ = b ddefsk Pos ~ ~ = b ) = sup{m( μ ~ ( x), μ ( x)) x r), (3.) ( ~ b Dlm keyty, ketk b ~ dlh sutu blg crsp (vrble) b, ddpt Pos{ ~ b} = sup{ μ { ~ ~ ( x) Pos < b} = sup{ μ~ ( x) Pos( ~ = b} = μ~ ( b) x R, x b} x R, x < b} (3.) Utuk f : R R r sutu opers deg blg ber dr hmpu fuzzy. Jk dotsk blg fuzzy ~, b ~ blg c ~ = f ( ~, b ~ ), mk fugs keggot μ c ~ dpt duruk dr fugs keggo μ ~ d μ b ~ deg Utuk sutu μ ~ ( z) = sup{m( μ~ ( x), μ~ ( y)) x, y R, z f ( x, y)} (3.3) c = b z R. Jd, posblstk bhw blg fuzzy c ~ = f ( ~, b ~ ) mempuy l z R dlh lebh besr dr kombs kemugk dr blg rl x,y sedemk z = f(x,y), dm l ~ d b ~ berturut-turut x d y. Secr umum, mbl f : R r sutu fugs deg l rl pd rug eucld - dmes. Jk utuk blg fuzzy ~, ~,..., ~ berk blg fuzzy c ~ = f ( ~, ~,..., ~ ), mk fugs keggot μ c ~ dlh duruk dr fugs keggot berkut: μ ~, μ ~,..., μ~ x r, =,,..., μ c~ ( z) = sup m μ ~ ( x ). (3.4) x z = f ( x, x,..., x ) Jd posblstk f ( ~ ~ ~,,..., ) b ddefsk Pos{ f ( ~ ~ ~,,..., ) b }= sup{ μ c~ ( z) z r, z b } (3.5) sebg

21 dm fugs keggot μ c ~ ddefsk pd (.4). Deg kt l, posblstk f ( ~ ~ ~,,..., ) b dberk Pos{ f ~ ~ ~ x, r, =,,..., (,..., ) b }= sup m μ ~ ( x ). (3.6) x f ( x, x,..., x) b Secr umum, sumsk bhw f : R r dlh fugs deg l rl pd rug eucld - dmes, =,,,m. Mk posblstk sutu sstem pertdksm f ( ~ ~ ~,,..., ) b, =,,3,..., m (3.7) dm b, =,,3,..., m dlh blg crsp (crspumber), ddefsk: Pos{ f ( ~, ~,..., ~ ) b, =,,..., m} = = sup x, x,... x m μ~ ( x ) f x x x b m (3.8) (,,..., ), =,,.., Pegert dr sstem pertdksm dlh byk kemugk ( r x, x,..., x ) utuk sstem pertdksm d l dr ~ yg bersoss deg Deg cr yg sm dperoleh, Pos{ f ( ~, ~,..., ~ ) b, =,,..., m} = d = sup x, x,..., x { m μ~ ( x ) f ( x, x,..., x) < b, =,,..., m} x, =,,...,. (3.9) Pos{ f ( ~, ~,..., ~ ) = b, =,,..., m} = = sup x, x,..., x m μ~ ( x ) f ( x, x,..., x ) = b, =,,..., m (3.) Jug deg cr yg sm betuk dr gbug perdksm d bersm.

22 3.4 Trpezodl blg fuzzy Sebg lustrs dberk deg umber fuzzy trpezodle, yg m dtetuk quttes fuzzy deg qudruple r, r, r, ) dr crsp umber sedek sehgg r < <, d membershp fucto: r r3 r4 ( 3 r4 x r, r x r r r, r x r3 μ ( x) = (3.) x r 4, r3 x r4 r3 r4, ltfel Kt ktk bhw fuzzy trpezodl dlh sutu blg fuzzy trughulr umber k r = r 3, dotsk deg trple ( r, r, r4 ). Ambl du blg fuzzy trpezodl ~ ~ r = ( r, r, r, ) d b = ( b, b, b, ), dtuuk pd gmbr 3. 3 r4 Jk r b3, mk dperoleh Pos = m 3 b4 { ~ ~ r b } = sup { m{ μ ~ r ( x ), μ ~ ( y )} x y} b { μ~ r ( r ), μ~ ( b ) } m 3 {, } =, b ~ Deg mplks pos { ~ r b} =. Jk r b3 ş r b4 mk supremum dlh ttk δ x yg merupk hsl rs dr du membershp fucto. Perhtugy dpt dlkuk deg megguk: d Pos { ~ ~ r b} = δ = ( b δ = r + r )δ x ( r 4 b4 r b ) + ( r 3 r )

23 μ ~ ( x) μ ~ ( x) b r δ b b r b 3 δ x r r 3 b 4 r 4 Fgur 3.: du blg fuzzy trpezodl ~ r ş b ~. Jk r > b4, mk utuk sutu x < y, slh su dr persm dpt dperoleh μ ~ ( x) =, μ~ ( y) = r Jd dberk Pos { r b} b ~ ~ =. Kt peroleh, r b3 ~ Pos{ ~ r b} = δ, r b3, r b4 (3.), r b4 Secr khususl, utuk b ~ dlh blg crsp, ktperoleh, r Pos{ ~ r } = δ, r r (3.3), r dm r δ = (3.4) r r

24 Dr ur d ts dpt dbuktk lem berkut. Lem. Sudrdt [5, 6, 7, 8] Berk blg fuzzy trpezodl r = ( r, r, r ) utuk cofdece level α deg α, { r } α ( α ) r + αr. ~ 3, r 4. Mk Pos ~ k d hy k Bukt. Jk Pos { ~ r } α, mk r tu r ( ) α. Jk r r r, mk r < r r sedemk sehgg ( α ) r + αr. Jk α, mk r α ( r r ) kre ( r r ) r <. Dr dperoleh α ) r + αr. r ( Jk ( α ) r + αr, dpt d urk dlm du ksus. Utuk r, dperoleh Pos { ~ r } =, megkbtk Pos { ~ r } α. Utuk r > dperoleh r r r < sehgg ( α ) r + αr tu α, deg ( r r ) kt l, Pos { ~ r } α. Lem terbukt. Dr opers ber (.3), kt peroleh umlh dr trpezodl fuzzy umber ~ = (,,, ) ~ ş b = ( b, b, b, ), dlh 3 b4 { m{ μ~ ( x), ( y z = x y} μ z ~ ~ ( ) = sup b μ~ )} + + b z ( + b ), + b z + b ( + b ) ( + b ), + b z 3 + b3 = z ( + b 4 4 ), 3 + b3 z 4 + b ( 3 + b3 ) ( 4 + b4 ), ly 4 3 4

25 Jk, umlh du trpezodle fuzzy umbers dlh sm deg trpezodl fuzzy umbers, d ),,, ( ~ ~ b b b b b = +. (3.5) Perkl trpezodl fuzzy umbers deg sklr. dlh { } x z x z μ μ = = ) ( sup ) ~( ~. meghslk < = ),,,, ( ),,,, (. ~ (3.6) Perkl trpezodl fuzzy umbers deg sutu sklr dlh stu trpezodl fuzzy umbers. Jumlh trpezodl fuzzy umbers dlh sm deg trpezodl fuzzy. Sebg cotoh, sumsk bhw ~ dlh trpezodle fuzzy umbers ),,, ( 4 3, d dlh blg sklr yg bersesu utuk,...,, =. Defsk = +,,, ltfel dc ( =,,, ltfel dc ( utuk,...,, =, mk + d dlh semuy oegtf d memeuh + =. Jumlh d perkl trpezodl fuzzy umbers, dperoleh T = = = + = + = + = + = ) ( ) ( ) ( ) (. ~ ~

26 LEMMA 3. Sudrdt [5] Asumsk bhw blg fuzzy trpezodl ~ r = ( r, r, r3, r 4 ). Mk utuk sutu cofdece level α yg dberk, α, Pos ( ~ r ) α k d hy kf ( α) r + α r. Hmpu level dr blg fuzzy ~ r = ( r, r, r3, r 4 ) dlh crsp subset dr R d dotsk ~ ] [ r = { x μ( x), x R}, deg megcu pd Crlsso dkk. [4], dperoleh ~ ] [ r = { x μ( x), x R} = [ r + ( r r ), r4 ( r4 r3 )]. Berk ~ ] [ r = [ ( ), ( )], l rt-rt crsp possblstk dr ~ r = ( r, r, r3, r 4 ) dlh ~ E ( ~ r ) ( ( ) + ( )) d, = dm E ~ meotsk fuzzy me opertor. ~ r = r, r, r r dlh trpezodl fuzzy umber mk Dpt dlht bhw k ( ) 3, 4 (3.8) ~ ( ~ r + r3 r + r E r ) = ( r ( ) ( )) 4 + r r + r4 r4 r3 d = +. Buktk! 3 6

27 BAB IV FUZZY LOGIC 4.. Fuzzy Logc Profesor Lotf A. Zdeh [35] dlh guru besr pd Uversty of Clfor yg merupk pecetus seklgus yg memsrk de tetg cr meksme pegolh tu meme ketdkpst yg kemud dkel deg logk fuzzy. Dlm peyy vbel-vrbel yg k dguk hrus cukup meggmbrk ke-fuzzy- tetp d l phk persm-persm yg dhslk dr vrble-vrbel tu hruslh cukup sederh sehgg komputsy med cukup mudh. Kre tu Profesor Lotf A Zdeh kemud memperoleh de utuk meyky deg meetuk dert keggot (membershp fucto) dr msg-msg vrbely. Fugs keggot (membershp fucto), Sudrdt [5] dlh sutu kurv yg meuukk pemet ttk put dt kedlm l keggoty (serg ug dsebut deg dert keggot) yg memlk tervl tr smp. Dert Keggot (membershp fucto) dlh : dert dm l crsp deg fugs keggot ( dr smp ), ug megcu sebg tgkt keggot, l keber, tu msuk fuzzy. Lbel dlh m deskrptf yg dguk utuk megdetfksk sebuh fugs keggot. Fugs Keggot dlh medefsk fuzzy set deg memetkk msuk crsp dr domy ke dert keggot.

28 Gmbr 4. Kosep dsr logk fuzzy Msuk Crsp dlh msuk yg tegs d tertetu. Lgkup/Dom dlh lebr fugs keggot. Jgku kosep, bsy blg, tempt dm fugs keggot dpetkk. Derh Bts Crsp dlh gku seluruh l yg dpt dplksk pd vrbel sstem. Pd tekk dgtl, Dubos d Prde [5], dkel du mcm logk ytu d sert tg opers dsr ytu NOT, AND d OR. Logk semcm dsebut deg crsp logc. Logk serg dperguk utuk megelompok sesutu hmpu. Sebg cotoh, k dkelompokk beberp mcm hew, ytu hu, kkp, pr, kucg, kmbg, ym ke dlm hmpu k. Sgt els bhw hu, kkp d pr dlh ggot hmpu k sedgk kucg, kmbg, ym dlh buk ggoty, sepert dtuuk pd Gmbr 4..

29 Gmbr 4.. Pegelompok beberp hew ke hmpu k Nmu kdg kl dtemu pegelompok yg tdk mudh. Mslk vrbel umur dbg med tg ktegor, ytu : Mud : umur < 35 thu Proby : 35 umur 55 thu Tu : umur > 55 thu Nl keggot secr grfs, hmpu mud, proby d tu dpt dlht pd Gmbr.3. Gmbr 4.3 Pegelompok umur ke hmpu ktegor us crsp logc Pd Gmbr 4.3 dpt dlht bhw : Apbl seseorg berus 34 thu, mk dktk mud (µ mud [34] = ) Apbl seseorg berus 35 thu, mk dktk tdk mud (µ mud [35] = )

30 Apbl seseorg berus 35 thu kurg hr, mk dktk tdk mud (µ mud [35th hr] = ) Apbl seseorg berus 35 thu, mk dktk proby (µ proby [35] = ) Apbl seseorg berus 34 thu, mk dktk tdk proby (µ proby [34] = ) Apbl seseorg berus 35 thu kurg hr, mk dktk tdk proby (µ proby [35th hr] = ) Dr s bs dktk bhw pemk hmpu crsp utuk meytk umur sgt tdk dl, dy perubh kecl s pd sutu l megkbtk perbed ktegor yg cukup sgfk. Hmpu fuzzy dguk utuk megtsps hl tersebut. 4. Serh perkembg fuzzy logc Fuzzy logc dlh cbg dr mtemtk deg btu computer memodelk du yt sepert yg dlkuk mus. Fuuzy logc meformulsk mslh memd lebh mudh, mempuy pess y tgg, d solus yg kurt. Fuzzy logc megguk dsr pedekt hukum-hukum utuk megotrol system deg btu model mtemtk. Pd Boole Logc setp petryt beru tu slh, se cotoh peryt deg tu. Jelsy hmpu fuzzy memlk fleksblt keggot yg dperluk utuk keggot pd sutu hmpu. Setp ked dr tgkt d ls yg els dlh meuuk ksus terbt pd pedekt yg ber. Kre tu dpt dsmpulk bhw Boole Logc dlh subset dr I Fuzzy Logc. Serh perkembg fuzzy logc sebg berkut: [37] 965 Pper pertm Fuzzy Logc oleh Prof. Lotf Zdeh, Fculty Electrcl Egeerg, U.C. Berkeley, sets the foudto stoe for the fuzzy Set Theory 97 Fuzzy Logc ppled corol Egeerg. 975 Jp mkes etry 98 Emprcl Verfcto of Fuzzy Logc Europe Brod Applcto of Fuzzy Logc Jp.

31 99 Brord Applcto of Fuzzy Logc Europe d Jp 995 U.S creses terest d reserch Fuzzy Logc. Fuzzy Logc becomes Stdrd Techology d s wdely ppled Busess d Fce. Gmbr 4.4. Fuzzy logc d Bole Logc Gmbr 4.4. b Fuzzy logc d Bole Logc Teor hmpu fuzzy merupk kergk mtemts yg dguk utuk mempesetsk ketdkpst, ketdkels, kekurg forms d keber prsl, Tettmz.

32 4. 3 Crsp Set d Fuzzy Hmpu Crsp (Crsp Set) A ddefsk oleh tem-tem yg d pd hmpu tu. Jk A, mk l yg berhubug deg dlh. Nmu, k A, mk l yg berhubug deg dlh. Nots A = { x P( x) } meuukk bhw A bers tem x deg P (x) ber. Jk X A merupk fugs krkterstk A d propert P, dpt dktk bhw P (x) ber, k d hy k X A ( x) =. Hmpu fuzzy (fuzzy set) ddsrk pd ggs utuk memperlus gku fugs krkterstk sedemk hgg fugs tersebut k meckup blg rel pd tervl [,]. Nl keggoty meuukk bhw sutu tem tdk hy berl ber tu slh. Nl meuukk slh, l meuukk ber, d msh d l-l yg terletk tr ber d slh. Seseorg dpt msuk dlm hmpu berbed, Mud d Proby, Proby d Tu. Seberp besr eksstesy dlm hmpu tersebut dpt dlht pd l keggoty. Gmbr 4.4 meuukk hmpu fuzzy utuk vrbel umur. Gmbr4.5 Grfk pegelompok umur ke hmpu ktegor us deg logk fuzzy

33 Pd Gmbr 4.5 dpt dlht bhw : Seseorg yg berumur 4 thu, termsuk dlm hmpu mud deg µ mud [4] =,5; mu umur tersebut ug termsuk dlm hmpu proby deg µ proby [4] =,5. Seseorg yg berumur 5 thu, termsuk dlm hmpu tu deg µ tu [5] =,5, mu umur tersebut ug termsuk dlm hmpu proby deg µ proby [5] =,5. Pd hmpu crsp, l keggoty hy d du kemugk, ytu tr tu, sedgk pd hmpu fuzzy l keggoty pd retg tr smp. Apbl x memlk l keggot fuzzy µ A [x] =, berrt x tdk med ggot hmpu A, ug pbl x memlk l keggot fuzzy µ A [x] = berrt x med ggot peuh pd hmpu A. Istlh fuzzy logc memlk berbg rt. Slh stu rt fuzzy logc dlh perlus crsp logc, sehgg dpt mempuy l tr smp. Perty yg k tmbul dlh, bgm deg opers NOT, AND d OR-y? Ad byk solus utuk mslh tersebut. Slh stuy dlh: - opers NOT x dperlus med - µ x, - x OR y dperlus med mx(µ x, µ y ) - x AND y dperlus med m(µ x, µ y ). Deg cr, opers dsr utuk crsp logc tetp sm. Sebg cotoh : - NOT = = - OR = mx (,) = - AND = m (,) =, d dperlus utuk logk fuzzy. Sebg cotoh : - NOT,7 =,7 =,3 -,3 OR, = mx (,3,,) -,8 AND,4 = m (,8,,4) =, Vlds d kosstes pd fuzzy logc

34 Notsk V, V, L, V vrbel logc.dlm fuzzy logc dsumsk bhw vrbel V deg l dlm tervl [,]. Defs 4. Negoţă [4]. Sutu vrbel V dlh formul fuzzy; b. Jk A dlh formul fuzzy, mk A (egs) dlh formul fuzzy; c. Jk A, A' dlh formulfuzzy, mk A A' (cous) d A A' (dsus) dlh formul fuzzy. Defs 4. Negoţă [4] Berk F (A) dlh l logc utuk formul fuzzy A. Asumsk bhw memeuh xom berkut:. A = V F A) = F( V ) ; ( b. F ( A A') = m( F( A), F( A' )) ; c. F ( A A') = mx( F( A), F( A' )) ; d. F( A) = F( A) ; Cotoh: F( V, V, L, V ) = m( F( V ), F( V ), L, F( V ) Defs 4.3 Negoţă [4] Sutu formul fuzzy A dsebut vld ( kosste), k F ( A) F( A) utuk semu td dr kemugk vrbel dr A. Sutu formul fuzzy A dsebut ovld (kosste) k tdk (kosste). Defs 4.4 Negoţă [4] Sutu formul A dktk betuk orml kougs, k A = P, P, L, P,, dm P dlh proposs fuzzy. Sutu formul A dktk betuk orml dsugs, k A = Φ Φ L Φ,, dm dlh fuzzy logc. Φ

35 Teorem 4. Negoţă [4 ] Sutu proposs P dlm fuzzy logc dlh vld, k d hy k P memut psg dr vrbel V, ). ( V Bukt. Jk P memut psg komplemeprbel V, ), dr defs P dpt dtuls dm P = L L L L, L lterry,kbty F( P) = mx{ F( L k m L = V tu L = V k ( V. Htug )} mx( F( V ), F( V )). Jk F ( V ), sehgg F ( P). Jk F ( V ), mk dr Defs 4. F ( V ), d utukksus F ( P), mk P dlh vld. Seblky, k P dlh vld, sumsk mellu bsurd, bhw P tdk memut psg ( V, V ). Mk sumsk sutu td utuk F( V k ), k =,, L,, oleh kre tu F ( V ). Sedemk sehgg F ( P). Kotrdks bhw, d teorem terbukt. Teorem 4. Negoţă Negoţă [4] F dlm logk dlh tdk kosste k d hy k F memut psg berurut vrbel V, ). ( V Bukt Deg cr yg sm pembukt dpt dlkuk sm sepert pd Teorem 4.. Cololr Negoţă [4] Sutu formul fuzzy A = P P L P dlm betuk ormlkougs ( A = Φ Φ L Φ dlm betuk orm dsugs) dlh vld (cosste), k d hy k semu { } dlh vld = Φ } dlh tdk kosste). P { =

36 Bukt Kods k { P} = dlh vld (dlm fuzzy logc), mk F ( P) utuk semu td yg mugk. Dr Defs 4., kt peroleh F ( A) = m{ F( P )} l d deg demk A dlh vld. Keblky, k F ( A), mk, F( P ) F( A) d deg demk } dlh vld. { P = Teorem 4.3 Negoţă [4] Sutu formul A dlh fuzzy vld(fuzzy cosste) k d hy k A hádl vld (cosste).

37 BAB V PEMOGAMAN LINIERR FUZZY 5.. Pemogrm ler fuzzy Pemrogrm ler dlh sutu cr utuk meetuk l optmum (mksmum tu mmum) dr sutu fugs ler dbwh kedl-kedl tertetu yg dytk dlm betuk persm tu pertdksm ler. Fugs ler yg dcr l optmumy tu dsebut fugs obektf tu fugs tuu. Betuk umum mslh pemrogrm ler dpt drumusk sebg berkut : mksmum( mum) : z = cx deg kedl : x b (5.) x, dm x ( L ) T = x,, x dlh vektor vrbel, c ( L ) T = c,, c dlh vektor by, A = ( ) dlh mtrks kedl berukur m d b ( L ) T = b,, b m dlh vektor rus k. Hmpu semu vektor x R yg memeuh semu kedl dsebut hmpu lyk. Betuk umum tersebut ug dpt dsk dlm betuk sebg berkut : mksmum( deg kedl : mum) : z = m = = x = x c x b, =, m,, =,. =, (5.) Dlm byk plks, fugs obektf mupu kedl-kedly sergkl tdk dpt dytk deg formul yg tegs tetp kbur. Oleh kre tu pemrogrm ler (tegs) dkembgk med pemrogrm ler kbur tu fuzzy ler progrmmg, deg betuk umum dlh sebg berkut :

38 Fuzzy pemogrm ler, dkemukk oleh Bellm d Zdeh [], dlh pegembg dr pemogrm ler (PL) deg fugs obectf d kedl dytk deg hmpu fuzzy sets. Defsk mslh LP deg crsp dr kedl fuzzy, d crsp tu obektf fuzzy dlh: ~ T mx Z = c X, m Kedl: ~ X b, =, p (5.3) = X, ~ dm sumber dy fuzzy resources b, sm deg fugs keggot (membershp fucto). Model pertdksm kedl fuzzy: ~ T mx Z = c X, m subect to ~ X b ~, =, p (5.4) = X, betuk (5.3) d (5.4) dlh merbed dlm beberp hl, utuk megts hl tersebut dpt megguk beberp pedekt dbwh pre-ssums dr fugs keggot dr fuzzy sumber yg tersed d fuzzy pertdksm kedl. Perbed tr crsp d kedl fuzzy dlh dlm ksus crsp kedl pegmbl keputus dpt medefereslk secr strp tr fesbel d ketdk fesbel; dlm ksus kedl fuzzy yg megdug tkt dr kefesbel dlm sutu tervl,werers [33]. Beberp pedekt pd model-model pemogrm ler fuzzy (Fuzzy Ler Progrmmg, FLP), Sudrdt [5] d Tk [3]. Pedekt pertm: Sumber dpt dtetuk secr pst, mslh LP trdsol: subect to mx Z = c m = X X, T X, b ~, (5.5)

39 dm c, d b, dlh derk secr tept. Solus optml (5.5) dlh uk. Pedekt kedu Chs d Verdegy [ pd 5]: Pegmbl keputus meghrpk dpt membut sutu lss postoptmzto. Jd mslh pemogrm prmetrc dpt dfomulsk: kedl mx Z = c m = X X, T X, ~ b + θ p, θ [,], (5.6) dm c,, b d p, dlh derk secr tept d θ dlh sutu prmeter,, p dlh tolers mxmum yg sellu postf. Solus Z * ( θ ) dr (5.6) dlh fugs dr θ. I dlh utuk sutu θ dpt dtetuk solus optml. Deg kt l, sumberdy hrus dlm betuk fuzzy. Mk mslh LP deg sumberdy fuzzy med: subect to mx Z = c m = X, b ~, X. Hl mugk utuk meetuk tolers mksmum X T (5.7) p dr sumber dy fuzzy b,. Kemud dpt d betuk fugs kegot μ dsumsk ler utuk sutu kedl fuzzy, sepert dwh : m f X b, = m X b m = μ = f b X b + p, (5.8) p = m f X > b + p. = μ [x] x

40 b b + p p Gmbr 5. Fugs keggot Dr Gmbr 5. dpt dlht bhw, semk besr l dom k memlk l keggot yg cederug semk kecl. Sehgg utuk mecr l -cut dpt dhtug sebg = t, deg : b + tp = rus k bts ke- Deg demk k dperoleh betuk ler progrmmg bru sebg berkut : mksmumk : deg kedl : p + x b + p, =, m x Chs d Verdegy, meytk bhw (5.7) d (5.8), ug ekvle deg (5.5), sutu LP prmetrc dm c,, d p, dlh dberk, deg megguk kosep -level cut. Utuk sutu -level cut pd hmpu kedl fuzzy (5.7) med mslh LP. Jd, mx Z = c T X, kedl X X, (5.9) X = { X μ,, d X, [,]. d equvlet to:

41 kedl ~ mx Z = c m = X b X, [,] d X, T + ( ) p,, (5.) dm c,, b, d p, dlh dberk deg tept. berkuty, Jk Hmpu = θ, mk persm (5.) k sm deg (5.6). Mk tbel solus dperlhtk kepd pegmbl keputus utuk meetuk kecukup solus. Z * ( θ ), θ [,] dlh solus fuzzy yg sesu deg pedekt Verdegy s pproch [ ]. Pedekt ketg (Weers s pproch [33]): Pegmbl keputus meyelesk mslh FLP deg fugs obectf fuzzy d kedl fuzzy, deg b, dlh tdk dkethu. Perhtk model: ~ mx Z = c T X, kedl m = ~ X X, b,, (5.) ekvle deg: kedl ~ mx Z = c = ~ X b X, [,] d X, T + θ p,, (5.) dm c,, b, d p, dkethu, tetp gol dr fugs obectf fuzzy dlh tdk dkethu. Peyeles (5.) de meguk pedekt Werers s [33], pertm defsk Z sepert d bwh : Z = f(mxc X X T X ) = Z * ( θ = ), (5.3) Z d Z = sup((mx c X X T X ) = Z * ( θ = ) (5.4)

42 dm X = { X X b + θ p,, θ [,], d X }. = Fugs keggot Werers s μ dr fugs obektf fuzzy, dlh: Fugs keggot T f c X > Z, T Z c X T μ = f Z c X Z, (5.5) Z Z T f c X < Z. μ,, dr kedl fuzzy dlh ddefsk pd (5.8). Deg megguk m-opertor dr Bellm d Zdeh [], dpt dtetuk rug keputus D yg ddefsk deg fugs kegot μ D dm, μ D = m( μ,..., μ p ). (5.6) I merupk pemlh keputus yg tept dm μ D dlh mksmum solus optml dr (5.). Mk dr tu (5.) ekvle deg: mx μ, μ,, μ d μ [,], X, (5.7) dm c,, b, d p, dlh dkethu, d = μ D = m( μ,..., μ m ). Ambl = θ. Mk mslh pd (5.7) ekvle deg: mx θ kedl T c X Z ( X ) b θ ( Z + θp,, θ [,] d X, Z ), (5.8) dm c,, b, d p, dlh dkethu d θ dlh bg dr ( Z Z ) utuk kedl pertm d bg tolers mksmum y ly. Solus dlh merupk sulus optml uk.

43 Pedekt keempt (Zmmerm s pproch [36]): Pegmbl keputus meggk peyeles mslh FLP deg fugs obectf d kedl fuzzy, dm gol b dr fugs obektf fuzzy d utuk toles mmum dlh dketehu. Hl, ~ T mx Z = c X, kedl = ~ X X, b + θ p,, (5.9) dm c,,, b, p b d p, dlh dkethu. Mslh yg dberk pd (5.9) dlh ekvle deg: Tetuk X, kedl ~ mx Z = c m = X X, T b X, + θ p,, (5.) deg fugs keggot dr kedl fuzzy yg dberk sebelumy pd (5.) d fugs keggot dr fugs obectf fuzzy μ sepert dbwh : T f c X > b, T b c X T μ = f b p c X b, (5.) Z Z T f c X < b. p Jd deg megguk kosep mksmum, (5.8) dlh ekvle deg: mx kedl μ d μ,,, μ d X, μ [,], (5.) dm,, b, p b d p c,, dlh dkethu,. berk = θ. Mk (5.7) k ekvle deg:

44 kedl mx θ, c T m = X b X + θ p b + θ p,, θ [,] d X,, (5.3) dm c,,, b, p b d p, dlh dkethu d θ dlh sutu bg dr tolers mksmum. Solus optml (5.3) dlh uk. Pd st fugs obektf fuzzy dsumsk, pedekt Zmmerm [36] d Werer s [33] dlh megsumsk petgy performce pd fugs obektf, T f ( X ) = F( c X ) [,]. * Mk, pd semu ksus, k Z ( θ ), θ [,], dlh solus fuzzy pd mslh, yg berkorespodes solus utuk setp fugs obekctf fuzzy (performce fucto ssocted to Zmmerm s or Werer s pproch) yg dpertmbgk. Pedekt kelm: Pegmbl keputus meyelesk mslh FLP deg fugs obektf fuzzy d kedl fuzzy, deg hy gol b dr fugs obektf fuzzy dlh dkethu, tetp tolers p tdk dkethu. Pehtk, ~ mx ~ T mx Z = c X, kedl = ~ X X, b,, (5.4) dm c,, b,, b d p, dlh dkethu, tetep p tdk dkethu. Deg p tdk dkethu, dpt dkethu bhw p k berd dlm dtr d p [, b Z b Z. Utuk sutu ], ddptk fugs keggot dr fugs obektf fuzzy pd (5.). Kre sstem mempuy produktvts yg tgg l fugs obektf k besr mk Z t = θ, buk berrt memberk dert postf dr keggot utuk yg lebh kecl d Z.

45 Perbed mslh deg pedekt Zmmerm s dlh bhw pd mslh tdk dber l wl p. Sedgk pd pedekt Zmmerm s l wl p, p [, b Z ] dberk. Pegmbl keputus memlh kembl p dr semu solus utuk hmpu yg memut p. Pd mslh Zmmerm, pegmbl keputus memlh kembl p dlh tept, kre solus yg dpeoleh merupk solus optml (5.4). 5. Iterctve pemogrm ler fuzzy Proses pegmbl keputus k lebh bk pbl dbrk d dselesk deg meguk teor hmpu fuzzy, bhk lebh bk dr teor precse pproches. Nmu pr pegmbl keputus hrus memlk pemhm yg bk tetg tur-tur teor hmpu fuzzy oleh kre tu proses terctve tr decso mker d decso process cukup bk utuk meyelesk mslh yg sedg dhdp. D hl tu berber merupk tekk fuzzy ler progrmmg Gsmov[6], Rommelfger [8] d Sd [9]. Seluty kosep problems oreted dlh merupk kosep yg sgt petg dlm meyelesk mslh yt. Dlm megplksk teor hmpu fuzzy, user depedet (terctve) d mslh yg dhdp, kosep oreted, flexblty d robustess deg tekk pemogrm ler k member hsl yg lebh bk. Pd pedekt Iterctve Fuzzy Ler Progrmmg (IFLP) Skw [], Skw d Y [] d Sudrdt [5]deg pegtegrs smetrs Zmmerm s [36], Werer s [33], Verdegy s d Chs s FLP [pd ] drcg d dperbhru utuk sstem pedukug keputus dlm meyelesk specfc dom dr sstem Ler Progrmmg (LP), L d Hwg []. L d Hwg megurk expert decso support system k memberk solus yg bervrs utuk byk ksus yg rumt. Sebuh sstem meghslk fuzzy-effcet deg solus yg sgt bk d fuzzy ug meghslk solus yg efse. Hl bs d bh pertmbg bg pr pembut

46 keputus d sgt mudh melkuk modfks. Pd khry seorg pegmbl keputus dpt melkuk perubh k membershp fucto dr sebuh sstem, Werer, [33]. Sebuh plks Fuzzy Ler Progrmmg dpt meyelesk sutu mslh deg cr yg terctve L d Hwg []. Pd lgkh wl, model fuzzy d modelk deg sebuh forms yg ddpt, dm seorg pembut keputus dpt meyedk forms tersebut tp tmbh by yg mhl. Sebky memhm terlebh dhulu compromse soluto bhw seorg pegmbl keputus bs mersk bhw froms berkuty bs dperoleh d bs dpertmbgk utuk meghslk sutu keputus deg membdgk secr ht-ht k keutug d by yg dguk. Dlm hl lgkh-lgkh compromse soluto ug dpt meghslk keputus yg bk. Prosedur yg bk mewrk sesutu bts yg pst d forms memproses kompoe yg relev d oleh kre tu by forms k bs dtek, Rommefger [8]. Eleme yg sgt petg yg bs mempegruh solus k mslh Fuzzy Ler Progrmmg dlh ke fuzzy- prmeter yg k dguk dlm sebuh model. Bgm prmeter dlm fuzzy geometry merupk pot yg sgt petg. Kre keberhsl sebuh solus tergtug pd keberhsl k sebuh model dr sebuh sstem. Sel tu, terctve cocept memberk proses pembelr tetg sebuh sstem d membut kekebs pskolog bg pembut keputus. Sel tu member l solus yg bk. Fktor yg bk dlm sebuh sstem d desg sstem yg hgh-productvty, bhk optmlss dberk oleh sstem. Sebuh sstem Iterctve Fuzzy Ler Progrmmg dpt member tegrtooreted, peyesu d pembelr deg mempertmbgk semu hl yg tdk mugk dr sebuh dom dr permslh sebuh Ler Progrmmg deg tegrs deg logk IF THEN. Metode Iterctve Fuzzy Ler Progrmmg sudh dpelr sek thu 98. Peelty dlh Bptstell d Ollero, Fb, Cbobu, d Stoc,Ollero, Arcl d Cmcho, Se, d Skw [, ], Slowsk [3], Werer d Zmmerm. Zmmerm meergk beberp teor umum tetg metode pemodel dr decso support system, d sstem cerds pd lgkug fuzzy. Ly megembgk terctve pproches

47 utuk meyelesk mslh Multple Crter Decso Mkg (MCDM), L d Hwg []. Adpu tuu dr sebuh solus k sebuh model dlh sebg berkut, byk vrs model yg dpt dpelr dr sebuh model Ler Progrmmg. Nmu studes dr Zmmerm, Chs, Werers, d Vedegy sgt efse utuk meyelesk model Ler Progrmmg deg megguk decso support utuk meyelesk mslh yt Algortm Iterctve pemogrm ler fuzzy Lgkh-lgkh Algortm Iterctve Fuzzy Ler Progrmmg dlh sebg berkut, L d Hwg [], d Sudrdt [5] Lgkh Selesk mslh pemogrm ler klsk deg metode smplex. Sebuh solus optml yg uk deg correspodg cosumed resorces dberk kepd pr pembut keputus. Lgkh Lkuk solus utuk meykk Decso mker?, pertmbgk ksus dbwh :. Jk solus meykk, cetk hsly.. Jk resource, utuk beberp dlh dle llu dreduks terhdp b, kembl ke lgkh. 3. k l dr resource yg d tdk cukup tept d beberp l tolers yg dhslk msh memugkk mk lkuk lss prmetk, d lkuk lgkh 3. Lgkh 3 Selesk permslh pemogrm ler prmetrk. Llu hsly dsmp pd sebuh tbel. Pd st bersm seldk persm berkut : * * Z = Z ( θ = ) d Z = Z ( θ = ). Lgkh 4

48 Lkuk solus yg mugk kemud smp pd sebuh tbel utuk meghslk keputus. Pertmbgk kemugk kods dbwh :. Jk solus yg dberk bk mk cetk hsly.. Jk resource, utuk beberp pbl l yg dhslk tdk memusk mk tukr deg p, llu kembl ke lgkh Jk l obektf msuk kl mk term sebg slh stu solus d lutk ke lgkh 5. Lgkh 5 Setelh mempertmbgk hsl pd tbel, keputus dpt dtetuk ytu b sebg hsl d l tolers p utuk meyelesk mslh smetrs Fuzzy Ler Progrmmg. Jk hsl keputus tdk sesu deg gol dr sebuh l obektve fuzzy lkuk lgkh 6, k b dberk mk lgsug lkuk lgkh 8. Lgkh 6 Peyeles mslh (5.7) dsrk megguk solus Werer s. Lgkh 7 Apbl Solus (5.8) memusk, pertmbgk kemugk kods dbwh :. Jk solus yg dberk memusk mk cetk hsly.. Jk user sudh medptk l tuuy mk ytk b sebg hsl d lutk ke lgkh Jk resource, utuk beberp l dlh dle mk kurg p (d gt llu kembl ke lgkh. 4. Jk k dpt dtolers, utuk beberp l tdk dpt dterm mk gt deg p d kembl ke lgkh 3. Lgkh 8 l p sgt meetuk utuk meghslk sebuh keputus, k seorg pegmbl keputus g lebh mespesfks l dr p, mk hrus dsedk sebuh tbel llu lutk ke lgkh 9, k l p tdk tersed mk lgsug ke lgkh. p )

49 Lgkh 9 Selesk mslh (5.3) deg megguk metode Zmmerm s. Lgkh Apkh solus (5.3) Memusk?. Jk memusk mk cetk hsly.. Jk user teryt medptk hsl yg lebh bk ( d dlm bts tolers y) mk berk l b sebg gol (d p ) d kembl ke lgkh Jk resource, utuk beberp l dlh dle mk lkuk ters pd b ( d gt p d kembl ke lgkh. 4. Jk l dpt dtolers, utuk beberp l tdk dpt dterm mk gt deg p d kembl ke lgkh 3. Lgkh Selesk mslh terkhr. Llu pggl lgkh 9 utuk meyelesk mslh (5.3) utuk set p s. Llu solus dsmp pd sebuh tbel. Lgkh Apkh solus yg dhslk sudh memusk? Jk y, cetk l solus d khr soluto prosecedure, seblky lutk ke lgkh 3. Lgkh 3 berty kepd decso mker utuk meyrg l p, llu kembl ke lgkh, sgt berls utuk meyk decso mker p pd thp, kre terdpt de yg bk utuk l p terlht pd gmbr 5.. Utuk megmplemetsk IFLP, hy membutuhk two soluto-fdg techques, metode smplex d metode prmetk. Oleh kre tu IFLP k sgt mudh dbut pemrogrm dlm sebuh PC.

50 Model Formulto Effcet Extreme soluto Compromse Soluto Locl Iformto Soluto Acceptble Yes Best Compromse STOP No Modfcto of membershp fucto Yes Locl cosequece No Gmbr 5. Flow chr Decso Support system,werer s [33]

51 BAB VI DASAR-DASAR PEMODELAN Mtemtk dlh bhs yg melmbgk sergk mk dr peryt yg g dsmpk, Hlm []. Smbol-smbol mtemtk bersft "rtfsl" yg bru memlk rt setelh sebuh mk dberk kepdy. Tp tu, mk mtemtk hy merupk kumpul smbol d rumus yg kerg k mk. Bhs mtemtk dlh bhs yg berush utuk meghlgk sft kbur, memuk, d emosol dr bhs verbl. Lmbg-lmbg dr mtemtk dbut secr rtfsl d dvdul yg merupk per yg berlku khusus sutu permlh yg sedg dk. Kelebh l mtemtk dpdg sebg bhs dlh mtemtk megembgk bhs umerk yg memugkk utuk melkuk pegukur secr kutttf, Hlm []. Jk megguk bhs verbl, mk hy dpt megtk bhw S A lebh ctk dr S B. Apbl g megethu seberp eksky dert kectky mk deg bhs verbl tdk dpt berbut p-p. Terkt deg ksus mk hrus berplg ke bhs mtemtk, yk deg megguk btu logk fuzzy sehgg dpt dkethu berp dert kectk seseorg. Bhs verbl hy mmpu megemukk peryt yg bersft kulttf. Sedgk mtemtk memlk sft kutttf, yk dpt memberk wb yg lebh bersft eksk yg memugkk peyeles mslh secr lebh cept d cermt. Mtemtk memugkk sutu lmu tu permslh dpt meglm perkembg dr thp kulttf ke kutttf. Perkembg merupk sutu hl yg mpertf bl meghedk dy predks d kotrol yg lebh tept d cermt dr sutu lmu. Beberp dspl kelmu, terutm lmu-lmu sosl, gk meglm kesukr dlm perkembg yg bersumber pd problem teks d pegukur. Pd dsry mtemtk

52 dperluk oleh semu dspl kelmu utuk megktk dy predks d kotrol dr lmu tersebut. Pemodel mtemtk merupk kbt dr peyeles permslh yg terd dlm kehdup sehr-hr yg dselesk megguk mtemtk. Mslh yt dlm kehdup bsy tmbul dlm betuk gel-gel yg belum els hkkty, msh hrus membug fktor-fktor yg tdk/kurg relev, mecr dt-dt d forms tmbh, llu meemuk hkkt mslh sebery. Lgkh dmk sebg megdetfks mslh. Lgkh seluty setelh megdetfks mslh, mk mellu beberp pedefs ddk peeremh mslh ke bhs lmbg, ytu mtemtk. Peeremh dsebut pemodel mtemtk. Setelh model mtemtk d, mk dcr lt yg dpt dguk utuk meyelesky. Pemodel lh yg med kuc dlm peerp mtemtk. Memodelk mslh ke dlm bhs mtemtk berrt meruk tu mewkl obek yg bermslh deg rels-rels mtemts. Istlh fktor dlm mslh med peubh tu vrbel dlm mtemtk. Pd hkkty, ker pemodel tdk l dlh bstrks dr mslh yt med mslh/ model mtemtk 6. Kosep dsr sstem Defs 6. (Sudrdt [3]) Kumpul dr eleme yg slg berhubug stu sm l utuk mecp sutu tuu. Kosep dsr sstem pertm kl dkembgk oleh Vo Bertlffy sektr thu 94. Sebuh system dpt durk deg mesepesfksk:. eleme-eleme. lgkug 3. struktur ter 4. struktur ekster Setp eleme terdr dr seumlh subsstem, sedgk subsstem terdr dr sub-subsstem. 6.. Sft dsr system - sstem terdr dr eleme-eleme yg membetuk sutu kestu system, - dy tuu d slg ketergtug,

53 - dy terks tr eleme, - megdug meksme trsforms, - memlk lgkug (lgkug substsl, eleme lgkug yg terbts yg med med perht dlm bhs. 6.. Perkembg kesstem 9 9 Bolog Orgss sebg Sutu Kestu/keseluruh Sstem teks lebh kompleks -Psr lebh kompettf - proyek lebh mhl Geers Sstem Teor Kestu buk sekedr Peumlh bg Ilmu-lmu sosl & sosolog Metodolog sstem egeerg Metodolog percg tertegrs utuk meghslk rcg yg optml SISTEM Gmbr 6. Perkembg kesstem 6. Pemodel Mtemtk Dews, relt d dmk perkembg yg terd dlm msyrkt semk cept d rumt. Ked tersebut merupk hmbt tetp seklgus ug merupk ttg bg seorg pegmbl keputus. Oleh kre tu, utuk medptk keputus yg terbk, seorg pegmbl keputus dpt secr cermt megus kompleksts tu d megembgk ltertve pemechy, slh stu yg dpt dlkuk lh deg pedekt lss kutttf.

54 Utuk meglss permslh dperluk kemmpu pemhm secr sstemtk. Pd umumy sutu sstem terdr dr berbg mcm eleme yg sgt kompleks, sehgg utuk lss perlu dsederhk deg l meugky ke dlm sutu betuk fugs mtemtk tu betuk bstrks l yg dsebut Model. Model mempuy du cr, ytu sft represets d bstrks. Sft represets dcermk oleh sutu pemet dr krkterstk sstem yt yg k dpelr. Dsebut bstrks kre dlm model terd trsforms krkterstk d kompleksts ked yg kogkrt ke dlm bstrks deg megguk symbol-smbol mtemtk. Pembut model bertuu utuk medeskrpsk tu meelsk sekumpul fkt d seluty megguk model tersebut sebg lt kofrms. Beberp defs dr model:. Model dlh peggmbr dr sutu mslh secr kutttf.. Hdy A. Th,model merupk represets dr sutu sstem yt. 3. Utuk memperlhtk pegruh fktor secr sgfk. Tuu dr model dlh mergk yg del dr sstem yg bersgkut, yg meckup hubug fugsol dtr kompoe-kompoey. Cotoh model perlku Kurt Lew B = f ( P, E). Pembut model sebery merupk se utuk megtur kesembg dr du tutu yg bertetg, ytu model dtutut gr model dbut sesederh mugk gr pemech yg dhrpk mudh dperoleh, model mudh utuk dkedlk d mudh utuk dkomuksk, sedgk d phk l dkehedk gr model megdug sebyk mugk sft-sft dr sstem yg dpelr deg mksud gr supy model td meghslk pemech yg medekt ked yg sebery. 6.. Keutug dr pemodel Beberp keutug dr pemodel mtemtk dlh: Pertm, deg model dpt dlkuk lss d percob dlm stus yg kompleks deg megubh-ubh l tu betuk rels tr vrbel yg tdk mugk dlkuk pd sstem yt. Kedu, model memberk peghemt dlm medeskrpsk sutu ked yt.

55 Ketg, peggu model dpt meghemt wktu, by, teg d sumber dy berhrg ly dlm lss permslh. Keempt, model dpt mefokusk perht lebh byk pd krkterstk yg petg dr mslh. Kosep Acto Reserch Stus Persol yt Pegu d pegembg mellu peerp Pegembg cr pemhm stus persol Pegembg metodolog pemech persol yg sesu deg stus persol Gmbr 6. Kosep Acto Reserch 6.. Klsfks model Model dpt dklsfksk berdsrk:. Tuu. Deskrptf Sutu model yg dbut deg tuu utuk meuukk feome tertetu (ms llu). b. Normtf Sutu model yg dguk utuk mecr wb. c. Predktf Sutu model yg dguk utuk memperkrk ked-ked yg k dtg.

56 . Represettf. Secr bstrk Sutu model yg dytk dlm smbolk (model Mtemtk). Smbolk (kutttf, kulttf). Verbl b. Secr fsk (mrket dr sutu proyek) 3. Sstem. Stts b. Dms c. Rel d. Abstrk 4. Solus. Alss Model yg berush mecr l optml secr mutlk y x x = + + (d rumusy) b. Heurstk (lgortm) c. Smuls Kemugk-kemugk dr solus dcr/dcob (mecr solus yg fesble) Meurut Russell L. Ackoff, scetfc Method,Sudrdt [3] model dpt dklsfksk sebg berkut:. Model Ikok Merupk vers mture, tetp sft-sft kesly tetp d. Model dguk kre kt g medptk sutu gmbr tetg sstem yt.. Model Alogk Pempl fsk berbed, tetp dpt memperlhtk perlku yg tetp sm. 3. Model Altk Sutu model yg memplk betuk fsky, bsy betuk model mtemtk tu logk. Meurut Wlso, sm deg meurut R.L. Ackof hy dtmbh deg Model Koseptul.

57 Dsr Klsfks. Fugs. Strukturl 3. Dmes 4. Tgkt kepst 5. Pegruh wktu 6. Tgkt Geerlss 7. Tgkt keterbuk 8. Tgkt Kutfks Klsfks Model. Model Deskrptf b. Model Predktf c. Model Normtf. Model Ikok b. Model Alogk c. Model Smbolk. Model du dmes b. Model tg dmes. Model Pst b. Model Koflk c. Model Resko d. Model tk pst. Model Stts b. Model Dms. Model Khusus b. Model Umum. Model Terbuk b. Model Tertutup. Model Kutttf b. Model Kulttf 6..3 Klsfks model ltk. Stedy stte Determstk Setp model yg megguk model lbr (prog.mt/lp) b. Stedy stte o determstc Dguk bl meksme perlku tdk dkethu, tetp dpt dsumsk dy vrbelvrbel yg secr totl tu prsl tergtug dr yg l. Y = + bx + bx +... (6.)

58 Cotoh: Metode Motecrlo, megevlus persol determstc deg megguk persol probblstc. c. Dmk Determstk Model berdsrk (melbtk) persm dfferesl. Cotoh: ds F = m F = m dt ds dt k = d. Dymc Probblstk Model bsy dguk k meksme legkp, perlku tdk dkethu. Smuls: vrbel rdom dguk (6.) Tbel 6. Klsfks model ltk Stedy Stte Dymk Determstk Persm (lbr) Persm Dfferesl No Determstk Probblstk Hubug-hubug sttstc & probblstc Smuls Ked 6..4 Krkterstk model yg bk Model yg bk mempuy krkterstk:. Sederh Smpel dlm formulsy d ug smple dlm utlsery.. Robus Memberk wb yg cukup kurt deg kods yg kt temuk. 3. Model tu hrus komplt (Comprehesf) Arty mecermk d mewkl bg dr sstem. 4. Bersft dtf Apbl kt k megdk perubh mk perubh tu dpt dtegrsk pd model.

59 5. Mudh utuk dkedlk(pegguy mudh) 6. Mudh utuk dkomuksk pd org l Proses Pegembg Model Proses pemodel dpt d lht pd gmbr berkut: Rel World Coceptulzto Pressure Boudry Ecoomy... Coceptul Model Symbolc Model p, q ( r s ), t, f, 3... Iterpretto Gmbr 6.3 Prose pegembg model, (Erque 995)

60 Gmbr 6.4 Proses pegembg model, Sudrdt[3]

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