Nature and Science, 5(1), 2007, Han and Xu, Multi-variable Grey Model based on Genetic Algorithm and its Application in Urban Water Consumption

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1 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo Ha Ya*, u Shguo School of Cvl & Hydraulc Egeerg,Dala Uversy of Techology, ggog, Gajgz, Dala, 64 Cha,haya78@63.com ABSTRACT: Owg o he fluece of ecoomy, populao, sadard of lvg ad so o, he urba waer cosumpo possesses cera characerscs of grey. As he epaso ad compleme of grey sysem (G(,)) model, he mul-varable grey model (G(,)) reveals he relaoshp of resrco ad smulao bewee varables. Geec algorhm possesses he whole opmal ad parallel characerscs. I hs paper, hrough usg he geec algorhm, he parameer q of G(,) model has bee opmzed, ad a mul-varable grey model (G(,,q)) based o geec algorhm has bee bul. The model has bee valdaed afer eamg he urba waer cosumpo Dala cy from 99 o 3. The resul dcaes ha he mul-varable grey model (G(,,q)) based o geec algorhm s beer ha G(,) model, ad he G(,) model s beer ha G(,). [Naure ad Scece. 7;5:8-6]. KEYWORDS: Grey sysem; G(,,q); geec algorhm; urba waer cosumpo. INTRODUCTION Wh he rapd developme of ecoomy, perssely crease of populao ad cosaly mproveme of sadard lvg, he urba waer demad has bee cosaly rsg, bu he amou of waer supply s lmed. The coflc bewee urba waer demad ad supply s gradually eacerbag, ad he seleme of urba waer shorage s he ausere challege of urbazao developme. The forecas of urba waer cosumpo s he premse ad basc o pla ad maage waer resources. The resuls of forecas drecly fluece he relably ad praccably of assgme decso-makg of waer resources sysem, ad also drecly fluece he susaable cosumpo of urba waer resources ad susaable developme of socal ecoomy []. A prese, here are may mehods o predc he urba waer cosumpo, such as he regress aalycal mehod, he epoe smoohess mehod, he rao of waer cosumpo mehod, he grey sysem forecas mehod, ad he arfcal eural ework (ANN) mehod. ANN mehod eeds log seres of daa, so s dffcul o predc due o he lack of hsorcal daa. The grey sysem heory akes uceray sysem as he sudy objec, such as small sample ad poor formao. Through he creao ad developme of paral kow formao, he valuable formao s pcked ou, so he operao behavor ad evolveme rule are correcly descrbed ad effecvely moored ad corolled. Pracce has proved he grey sysem model eeds less formao bu he precso of resul s beer, ad ca preferably reflec he praccal codo of he sysem []. The grey sysem has bee eesvely appled producg, egeerg, scece ad echology. Owg o he fluece of ecoomy, populao, lvg sadard ad so o, he urba waer cosumpo has cera grey characerscs. Especally whe he loger seres of relable daa are uavalable, he grey sysem model s a avalable mehod o predc he urba waer cosumpo [3]

2 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo. UTI-VARIABE GREY ODE G(,) [4] Grey sysem model G(,) s dsposed hrough oe accumulao of sgle varable me seres { } (,,, ). Through frs-order dffereal equao, he rsc rule of geerag sequece ca be revealed. The G(,) ca oly be appled o modelg ad predcg of sgle me seres daa. d d + a b Oly usg he sgle me seres daa, he grey sysem model G(,)ca o reflec he fluece ad developme bewee each oher; however, he G (,) model ca maly be appled o descrbg he correlao of varable bewee each oher, o o predcg. The G (,) model has bee dealed descrbed referece [5]. I hs paper, he ul-varable Grey odel G(,) had bee roduced. G(,) model s varables frs-order dffereal equaos, s he aural epaso of G(,) model varables, o smple combao of G(,) model. The frs-order dffereal equao of G(,) model ca be wre as follows: d d d d d d a a a + a + a + a + + a + + a + + a + b + b + b The frs-order dffereal equao of G(,) model s as follows: d d A + B or A B d d The sequece me respose formula s () (3) A( ) A( ) ( ) e + A ( e I) B (,,, ) (4) Where A A e I A I ! k k A k! k The parameer A ad B ca be esmaed by leas-square mehod: T ( ) a ˆ bˆ T Y (,,, ) (5) - 9 -

3 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo Aˆ, ad bˆ b ˆ Bˆ bˆ where ( () + ) ( (3) + ()) ( + ( m )) ( ( ( () + (3) + + ) ()) ( m )) ( ( ( () + (3) + + ) ()) ( m )) ad Y ( (), (3),, ) T 3. THE ESTABISHENT OF G(,,q) ODE Sce orgal daa are maly me-seres daa pracce, he dervave ca be rasformed o a forward dfferece equao [6], such as A B I fac,, so we ca ge A B (6) + he same way, a backward dfferece equao s A B + A + B or (7) Dffere sequece sasfes dffere equao. Some sasfy he Eq.(6), whle ohers sasfy he Eq.(7). The commo dfferece equao s (8) + B + qa + ( q) A + The Eq.(8) s he commo dfferece equao of G(,). The buldg model s G(,,q) by Eq.(8). Whe q.5, he G(,,q) model urs o he G(,) model. For he ay q he backgroud mar s ( q () + ( q ) ) ( q (3) + ( q) ()) ( q + ( q) ( m )) ( q ( q ( q () + ( q ) (3) + ( q ) + ( q ) ) ()) ( m )) - -

4 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo ( q () + ( q ) ) ( q (3) + ( q ) ()) ( q + ( q ) ( m )) (9) T T ( ) Y,,,, a ˆ bˆ Accordg o he above, f oly gve q, he  ad Bˆ may be obaed by Eq.(9) ad Eq., he he may be obaed by Eq.(4). Whe he orgal seres are gve, he parameer s he oly facor ha flueces he precso of G(,) model. The relaoshp s very o-lear bewee q ad errors. If he geec algorhm s adoped, a deal value abou he parameer q may be obaed. I hs paper, he G (,) model s combed wh he geec algorhm (GA), whch s called G (,,q) model. q 4. THE SOUTION OF G(,,q) ODE WITH GA 4. The geec algorhm The geec algorhm(ga) [7,8] s a adapve whole search ad probably opmzao arhmec, whch smulaes he heredy ad evoluo of bology evromes. Ths arhmec was ally preseed by Joh Hollad, professor of chga Uversy U.S Through geec operao of seleco, ad muao o curre populao, he ew geerao s creaed ad gradually evolves o opmal sae. Oly he evaluao fuco s used durg seekg opmzao ad he dffereably of objecve fuco s o requred, ca he geec algorhm be whole, parallel, speed, adapable ad robus, so has bee eesvely appled he feld of fuco opmzao, produco corol, auomao corol, mage dsposal, arfcal lfe ad so o. 4. The basc seps of GA q [,] ) Ecodg. The ca be epressed by a bary cluser. The legh of (chromosome ) ca be deermed by he precso of q. q ) Ialzao of he populao. N umbers seleced from o a radom are regarded as al populao. (N s he umber of populao). 3) The fess evaluao of dvdual. The fess fuco dcaes he degree of adapao capably o he evrome, whch s relaed wh objecve fuco. The fess value of he No. dvdual - -

5 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo F (, (k s he erave mes) may be calculaed by usg he followg formula. C F(, ma f(,, f(, p C, f(, C The objecve fuco ca be epressed by he square sum of error; ma ma f(, N ( ˆ ).where C ma may be he mpor parameer, he mamum of f(, ul ow or he mamum of f(, a curre populao or laes several geerao populaos. 4) Seleco. The survval probably of he dvdual he k geerao s p ( k ) N F(, j F( j, (k ) We ca choose a sraegy (such as roulee), so he seleced probably of he dvdual s. The hgher fess of dvdual, he more chace of he dvdual ca be seleced as he ew dvdual. The lower fess of he dvdual, he fewer chace of dvdual ca be seleced as he ew dvdual, ad he more probably of beg elmaed. p 5) Crossover. A par of dvduals for crossover s seleced sochascally. The smples mehod of crossover s o selec a rucao po sochascally, spl each gee cha o wo secos a hs po, ad he echage her als, for sace: The crossover embodes he process of echagg formao course of bology heredy. 6) uao. Several dvduals are seleced from he populao wh probably ( ). For he seleced dvdual, a b s seleced radomly for muao, amely urs o (or o ). For eample [] s chaged o. The muao embodes he accdeal gee muao he course of bology heredy. p m 7) Evoluoary erao. The flal geerao from he prevous s regarded as he ew oe, he repea he 3) sep o 6) sep ul a accepable soluo wll be foud or he reserved erao mes fulflled. Through usg he geec algorhm, he deal ca be obaed; he mar ca be obaed by q - -

6 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo pug he q o Eq.(9); he he value of  ad Bˆ ca also be obaed by he Eq.. The calculao process of G(,,q) model s ˆ e A( ˆ ) Aˆ ( A e ) ( ) + ˆ ( I) Bˆ (,,, ) ˆ ( ) ˆ ˆ ( ) ˆ ( ) ( k) ( k) ( k ) (k,3, ) The relave error ad he square sum of errors ca be used o evaluae he G(,,q) model forecasg. The relave error s ˆ ; he square sum of errors s. ( ˆ ) 5. CASE STUDY Owg o he fluece of ecoomy, populao, sadard of lvg ad so o, he urba waer cosumpo possesses cera characerscs of grey. Due o he eed of less formao ad hgher precso, he grey sysem ca preferably reflecs sysem praccal codo. The followg s a aalyss ad calculao of urba waer cosumpo for several years Dala cy. Due o he avalable daa of waer cosumpo are scarce Dala cy, oly he daa from 99 o 3 ca be aalyzed Table. The Sasc of urba waer cosumpo from 99 o 3 Dala Year Urba waer cosumpo ( 4 m 3 ) Urba populao ( 4 people)

7 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo The daa from 99 o year are regarded as he basc, ad he daa from o 3 as he es. Accordg o he above heory, we perform a smulao forecas. I course of geec algorhm, he bary codg has bee adoped, he umber of populao s, he legh of codg s, he probably of crossover s.95, ad he probably of muao s.8. Afer opmzao of geec algorhm, he parameer q A las, we compare he smulao value ad he errors of G(,), G(,) ad G(,,q). The resul s show able. Table. The aalyss of smulao value ad errors abou hree models waer G(,) G(,) G(,,q) year cosump Smulao Relave Smulao Relave Smulao Relave o value errors value errors value errors ( 4 m 3 ) ( 4 m 3 ) (%) ( 4 m 3 ) (%) ( 4 m 3 ) (%) ea relave error(%) From he resul able, we ca draw he cocluso ha he precso of G (, ) s much hgher ha he G (,) due o he resrco ad smulao bewee varables. The G (,,q) s hgher ha he G (,) due o he parameer q beg wholly opmzed. The mea relave error s ehaced from 7.33% o 3.99%. The curves of modelg ad forecas of hree models are show Fgure

8 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo waer cosumpo year Fgure. The curves of smulao ad forecas of models real daa G(,) G(,) G(,,q) 6. CONCUSIONS As he epaso ad compleme of G (,) model, he G (,) ca reflec he relaoshp of resrco ad smulao bewee varables. Accordg o he characerscs of whole search ad cooave parallel calculao, he geec algorhm s adoped ad combed wh he mul-varable grey model (G (,) ) well, ad a ul-varable model (G (,,q)) based o Geec Algorhm has bee esablshed. Owg o he fluece of ecoomy, populao, sadard of lvg ad so o, he urba waer cosumpo possesses cera characerscs of grey. The grey sysem model s a avalable mehod o predc he urba waer cosumpo. Takg he daa of urba waer cosumpo Dala cy from 99 o 3 as eample, he model s proved. The resul dcaes ha he G(,,q) model s beer ha G(,) model, ad he G(,) model s beer ha G(,). A he same me, he course of modelg of mul-varable grey model, f he choce of varable s mproper, he morbdy of mar or verse mar would occur. How o choose he proper varable s dffcul he course of applyg, whch eeds o be furher researched. Correspodece o: Ha Ya School of Cvl & Hydraulc Egeerg Dala Uversy of Techology ggog, Gajgz, Dala, aog 64 Cha Telephoe: Emal: haya78@63.com Receved: arch 3, 7-5 -

9 Naure ad Scece, 5, 7, Ha ad u, ul-varable Grey odel based o Geec Algorhm ad s Applcao Urba Waer Cosumpo REFERENCES. I, ZUO Q-g. Comparave research o predcg mehod ad applcao for cy waer cosumpo [J]. Joural of Waer Resources & Waer Egeerg, 5, 6(3):6-. IU S-feg, DANG Yao-guo, ec, FANG Zh-geg. Theory ad applcao of grey sysem []. BejgScece Press, OU Yu, ZHANG Qg, IU Guo-hua, ec. Applcao of grey mul-varable forecasg model predcg urba waer cosumpo [J]. Waer Resources Proeco, 5, : ZHAI Ju, SHENG Ja-mg, FENG Yg-ju. The grey odel G(,) ad Is Applcao [J]. Sysems Egeerg Theory & Pracce, 997(5): DENG Ju-log. Grey Sysem Theory []. Wuha: Huazhog Uversy of Scece ad Techology Press, IE Ka-gu, I Chu-ya, ZHOU Ja-q. Grey model (G(,,λ) based o geec algorhm [J]. Joural of Sysems Egeerg,, 5(): Hollad J H. Geec algorhm[j]. Scefc Amerca,99(4): YAO We-ju. Sudy o back propagao ework opmzao wh geec algorhms [J]. Joural of Wuha Isue of Chemcal Techology, 4, 6(3):