INFORMATION RECYCLING MATHEMATICAL METHODS FOR PROTEAN SYSTEMS: A PATH-WAY APPROACH TO A GEOMETRIC PROGRAM. S. Kumar 1 ABSTRACT OPSOMMING

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1 hp://sje.jourls.c.z INFORMATION REYLING MATHEMATIAL METHODS FOR PROTEAN SYSTEMS: A PATH-WAY APPROAH TO A GEOMETRI PROGRAM S. Kumr Deprme of Memcs d Sscs Uversy of Melboure Ausrl sumr@ms.umelb.edu.u ABSTRAT I s ssumed pu vlues memcl model my chge frequely due o ercos my be eerl or erl d/or my be due o desg cosderos. Ths chgg evrome s ypcl desg suo where vrous possbles re red ou before ccepg oe s e fl desg. W e chgg evrome soluo procedure hs bee descrbed emps o ob e soluo o e chged problem by usg e formo d resuls lredy vlble from e soluo of e problem ws obed before chges occurred. The uor [5] hs clled s p-wy pproch d erms ll such meods bsed o s phlosophy s formo recyclg memcl meods. I s pper geomerc progrm (GP) s cosdered uder proe evrome. OPSOMMING De sewrdes v wsudge model verder dwels s gevolg v ere of esere erses e/of owerppssgs. Al e dwels vd d pls weer de owerpomgewg veres d versee owerpe oorweeg moe word voord fle beslu gem word. Vr herde suse word oplossg besryf w red hou me de mleu v verderg. De oueur besryf de meode v de loog v voepd -bederg e doop de flosfe me de elege m wsude slese lggsmeodes. Geomerese progrmmerg word gesp proeusgge omgewg. Also School of ompuer Scece d Memcs Vcor Uversy Fooscry Pr mpus PO Bo 8 Melboure y M 8 Ausrl. SA Jourl of Idusrl Egeerg Nov 6 Vol 7() 7-

2 hp://sje.jourls.c.z. INTRODUTION The cocep of developg formo recyclg meods ws descrbed d developed by Kumr [5]. I ose ppers he cosdered e cse of ler progrm qudrc progrm d secod order DE d developed lgormc dels for ech cse. Tluder d Kumr [6] d Tluder Kumr d Joes [7] developed e formo recyclg pproch for ewor problems rsg coeco w crcus free of specfed ls. Three erms re cerl o ese developmes whch re epled del [5]. These erms re: recyclg proe sysem d pwy pproch. Sce ey re essel for udersdg e pproch dscussed s pper bref eplo of ech s gve below. The recyclg cocep wse mgeme (Grover [7]) dels w ssgg wse from oe suo s resource o some oer suo. Smlr emps c be mde e memcl lyss. The old vlble formo fer model hs bee lysed s wse sce model s o loger vld represeo of e suo ws eded for. Ths hppes due o chges d cosequely vrous pu vlues lso chge. The recyclg cocep from wse mgeme moves oe o ulse e vlble formo s free resource for lyss of e chged suo f possble. Ths phlosophy s lbelled e formo recyclg pproch. The cocep of formo recyclg pproch s ppropre proe evrome [] where pu o model chges due o eerl d erl ercos. However e cse of geomerc progrmmg models whch re suble for desg problems chges lso rse due o desg cosderos. The repeed use of memcl model w chged pu vlues hs bee clled proe evrome by e uor s dscussed []. The objecve s pper s o develop formo recyclg meod for geomerc progrmmg model proe evrome. Flly e erm pwy pproch hs bee used. Usully soluo meods cree d of p from defed srg po order o rech e desred ed po. However e prese coe e defed srg po s e soluo obed before chges d e desred ed po s e requred soluo fer chges hve occurred e vlues of e pu prmeers. Thus p hs o be creed jos ese wo soluos where e srg po s e soluo before chges d e erml po s e soluo o e requred ew problem fer chges (see Grc d Zgwll [6]). Ths pper dels w geomerc progrm model s solved by usg e formo recyclg phlosophy. I s ow geomerc progrmmg model s ppropre for resolvg e vlues of e desg prmeers; d ccepg fl desg s mpor e suo s suded uder dffere sceros order o cosder vrous possbles. Thus repeed pplco of e sme model uder dffere pu prmeers s possble gve suo. So e sudy of geomerc progrm uder proe evrome s desrble d s pper s 8

3 9 emp o do. I seco bref dscusso of geomerc progrmmg model d codos of opmly s offered. A proe GP s preseed seco. The formo recyclg pproch for geomerc progrm hs bee developed seco d lgormc dels hve bee provded. Seco 5 dels w umercl llusros.. A GEOMETRI PROGRAM MODEL The opmly codos for GP model re summrsed s seco. Le GP model be gve by: Mmze ) ( X g where. > > () Thus ll re >. The rmec-geomerc equly s gve by: } m{ } m{ () where. > () Alervely relo () s equvle o: ( ) ( ) D D D () where j D j j (5) For e RHS of relo () o s mmum vlue e codos of opmly re: hp://sje.jourls.c.z

4 hp://sje.jourls.c.z D j j Thus relo () whe codos (6) re ssfed becomes ( ) ( ) (6) (7) Noe e LHS of (7) e uows o be deermed re d α j j re e ow gve vlues; d e uows e RHS fuco o be deermed re. These uow vlues c be deermed from e umber of codos gve by relo (6) d oe more codo s gve by e relo (). I ddo ll ese vlues re resrced o o-egve vlues. Thus f e umber of uows o be deermed e suo s lbelled s zero order of dffculy d whe > e problem s sd o hve ( ) order of dffculy. Whe ( < ) e problem s sd o be over-cosred. A deled ccou of GP model c be see [].. PROTEAN GP MODEL Sce vlues of e ow prmeers d j ; j wo cosecuve cosderos re ulely o rem e sme e vlues of uows RHS of (7) wll chge. The problem s o fd ew vlues of. d cosequely e ew vlues of e vrbles. I s seco pwy pproch for proe GP model s dscussed. All prmerc vlues re revewed for ech desg scero d ese sceros do chge for vrous cosderos d possbles. Oe my be eresed evlug ese dffere vlues uder dffere cosderos.. Opmly codos for e GP model wo cosecuve ervls To dsgush codos for wo cosecuve desgs e wo ses of codos for e desg d e ( ) desg re represeed by usg e followg oo. D j l l j. j j >. (8)

5 hp://sje.jourls.c.z d D j l j j l j. >. (9) Here e superscrp o e lef of symbol represes e desg scero or e desg umber. I s ssumed opml soluo o e model represeed by () for e desg hs bee obed by usg y ppropre soluo procedure []. The soluo for codos (9) requres eo oly f some or ll vlues of e vrous elemes j hve chged for e ( ) desg. Thus e problem s o fd soluo o codos (9) from e soluo lredy obed o codos (8) oer words o fd e p jos e ow po represeed by e soluo of codos (8) o ew po represes e soluo of codos (9). Also oe ese codos re depede of vlues. Oce soluo o. s ow oe hs o fd e correspodg ew soluo for e vrbles where vlues ply er role.. A ree-sge pproch o fd e soluo o e problem posed seco. I e followg e words desg d ervl re used erchgebly. They represe perodc revew d observo f pu vlues hve chged e reso beg erco or desg scero. I s ssumed e memcl problem for e ( ) ervl sll s GP model; chges rse oly w respec o e vlues of some or ll s elemes.e. vlues of j mgh chge. Ths s relsc ssumpo s vrous cosderos re lely o erc dffere wys. However s cer e chged vlues c be epressed by smple ddo d subrco w respec o prevous vlues of ose prmeers. Thus we c rewre ew vlues of e elemes j s follows: j j ; j j () Here ; j d c j j re uresrced ques.e. posve egve or zero ow vlues. To smplfy preseo of e bove dscussed des s ssumed wo cosecuve perods e umber of uow vrbles o be deermed rems uchged. Thus codos (9) c be epressed wo prs. I s ew preseo e pr oe codos re decl

6 hp://sje.jourls.c.z o (8) d e remg resdul vlues form e pr wo o e modfcos o ccou f y. Thus pr wo s epeced o be sprse srucure. Thus usg relos () codos (9) c be rewre s: D j l j. ( l j >. ) j j () Thus w respec o e desg codos (8) ddol sprse srucure hs bee dded o ose codos s show (). Here for smplcy hs bee ssumed e umber of vrbles d e umber of erms rem e sme wo cosecuve desgs.e. d e e umber of uow ' s o be deermed rem e sme s ws e cse for e cosdero. Iformo from e desg c be used o e o-zero colums of () o ob e correspodg equvle mr for e ( ) ervl model. Sce codos () or equvlely () re ler LP erpreo s possble w objecve fuco comprsed of e rfcl vrbles. The pproch s smlr o oe epled by e uor erler pper [5]. However whe colums of j d j re combed o oe colum ey my o be ble o hold o o s srucure w respec o LP ereme po erpreo. Furermore e fesbly d e opmly codos w respec o ( ) desg problem my o rem ssfed. I oer words s emphszed e resulg soluo w respec o e ( ) desg my o eve be ereme po soluo d my o eve be fesble. A s sge ree sge clculos o resore ese LP chrcerscs re used s follows: Sge Perform pvog operos o resore e bss.e. for ech bss vrble e desg u colum prevls for e ew d w respec o e ( ) desg. Sge Afer sge clculos f opmly codos re o ssfed use e smple eros o esblsh opmly codos. Sge I e resulg bleu fer sge compuos f e soluo does o ssfy e fesbly requreme use e dul smple meod o ob fesble soluo holdg o o e opmly codos.

7 hp://sje.jourls.c.z The ed of sge resuls ew soluo o vlues of. If ese vlues re > s e requred ew soluo for e ( ) desg cosdero. Recll ws ssumed d whch s o essel requreme for e proposed meod; ws jus smplfco for preseg e ree-sge clculos prcple. I e cse whe oe or bo codos re o ssfed oe s requred o crry ou more modfcos s epled subsequely e lgormc seps of e proposed meod. I my lso be oed ech bleu e ree sge clculos represes po dmesol spce whch whe joed o ech oer form pece-wse ler p jos e soluo of e problem for e cosdero o e requred soluo for e ( ) cosdero.. The Algorm The seps of e lgorm re s follows: Sep Record from e desg model vlues of: ; ; ; j ; j d e vlues of >.. Record e problem d for e ( ) desg cosdero.e. vlues of: ; ; ; j Sep ; j hec wheer e umber of vrbles for e cosdero s decl o e umber of vrbles for e ( ) cosdero; f so go o sep. Oerwse crry ou e followg modfcos: If vrble s o loger relev desg c be esly ccoued for by usg relo (); bu GP model dels w desg problem where ll vrbles re srcly requred o be >. The suo for () my rse oly whe e desg hs o be chged w respec o prculr dmeso. I cse we wre: j j () A more lely chge s ew vrble s requred for e ( ) cosdero compred o cosdero whe ew erm s dded o e LHS of problem (). I s suo e ew erm s smply dded o e sprse problem d subsued:

8 hp://sje.jourls.c.z () Sep ompre codos for uow for e d ( ) cosdero. If e umber of erms d umber of vrbles re equl for wo coseque cosderos e go o sep. Oerwse dd ew ( ) row d colum o esg X mr. The row s gve by: D l l l j. >. () Sep Rerrge e problem for e ( ) desg wo prs s show relo (). If e sprse pr of e problem does o es e soluo for e cosdero rems vld for e ( ) cosdero. Go o sep 6; oerwse go o sep. Sep W respec o e bss of e model for e cosdero perform pvog clculos o ech colum of e sprse problem d combe e correspodg colum j w j. rry ou pvo operos so e bss w respec o e desg s resored. If e resulg soluo does o ssfy e fesbly codo use dul smple eros o resore e fesbly of e ew soluo. Whe e fesble soluo hs bee obed go o sep 5. Sep 5 Sce bsc fesble soluo hs bee obed crry ou smple eros o ob opml soluo w respec o e rfcl vrble objecve. Go o sep 6. Sep 6 The curre soluo s fesble for e ( ) cosdero. Pr e soluo d ssg d go o Sep for e e ero.

9 hp://sje.jourls.c.z. PROTEAN GP MODELS: ILLUSTRATIVE EXAMPLES Emple The purpose of s emple s o llusre e lgorm o proe GP model seco where pu d chges for y reso. osder e followg GP model for e ervl. Mmze c c c c where c j > j. > (5) Le e bove be represeed s: c c c c (6) The e GP equly gves: c Mmum ( ) mmum c c c (7) The codos for opmly re: (8) Ths zero degree dffculy problem hs soluo gve by.5..5 whch hs bee obed from (8) gve by / 5 / 5 / 5 / 5. Assume s soluo of relos (8) ws obed by usg e LP pproch. Vrbles re represeed by X X X d X for e phse oe clculos whe four rfcl vrbles A A A A were dded o relos (8). The frs d ls ble of e smple ero s gve below: 5

10 6 Row X X X X A A A A RHS Obj A A - A - Row - Tble : Il formo w respec o (8) Row X X X X A A A A RHS X /5 9/5 /5 7/5 /5 X /5 - /5 /5 -/5 /5 X /5 /5 -/5-8/5 /5 X /5 -/5-6/5 /5 /5 Tble : Fl ble gvg vlues of requred vrbles d verse of e bsc mr Now cosder e problem e ) ( ervl become c c c c (9) Thus ew codos for vlues of > f e soluo ess re: () Rewrg () e wo prs by usg e relo () we ge Mmze ) ( A A A A Subjec o ) ( hp://sje.jourls.c.z

11 hp://sje.jourls.c.z () orrespodg o vrbles relos () e verse mr from Tble s ow. We use s verse mr o fd e correspodg colums for ew vrbles by usg e relo: B A j () The vlue of B s gve below: A A A A /5 9/5 /5 7/5 /5 - /5 -/5 /5 /5 /5 -/5-8/5 /5 -/5-6/5 /5 Tble : The verse mr elemes From () colum elemes w respec o four ew vrbles re respecvely gve by: d Usg relo () Tble d colum vecors () we ge e correspodg colum vecors s gve below (): 7 / 5 / 5 8 / 5 / 5 7 / 5 7 / 5 / 5 / 5 d 8 / 5 8 / 5 / 5 / 5 Thus ew vlues wll be s show Tble. () 8 / 5 / 5. () 8 / 5 / 5 Row X X X X X X X X A A A A RHS X 7/5 7/5-7/5-8/5 /5 9/5 /5 7/5 /5 X -/5 -/5 /5 /5 /5 -/5 /5 -/5 /5 X -8/5-8/5 8/5-8/5 /5 /5 -/5-8/5 /5 X /5 /5 -/5 /5 /5 -/5-6/5 /5 /5 Tble : Vlues w respec o e ( ) ervl problem. 7

12 hp://sje.jourls.c.z ombg colums of Xl d Xl for l d we ob Tble 5 gve below: New Vlue XX XX XX XX A A A A RHS X /5 7/5-7/5-8/5 /5 9/5 /5 7/5 /5 X -/5 /5 /5 /5 /5 -/5 /5 -/5 /5 X -8/5-8/5 /5-8/5 /5 /5 -/5-8/5 /5 X /5 /5 -/5 7/5 /5 -/5-6/5 /5 /5 Tble 5: Where X Xl l represe ew vrble Xl for e ( ) ervl Sce we hve los ll u colum srucure w respec o e vrbles X X X d X frs ese four colums re resored o e u colum srucure by usg e elemery operos o elemes of Tble 5 d w rerrgeme s s gve Tble 6. New Vlue XX XX XX XX A A A A RHS X / 6/ -/ 9/ / X / -9/ 5/ -/ / X / / -/ -8/ / X 7/ -/ -/ / 7/ Tble 6: Resored equvle vlues from Tble 5. Sce ll fesbly d opmly codos re ssfed e resulg ew soluo for e ( ) cosdero s gve by X / X / X / X 7 /. Oce ese ew vlues re ow oe s requred o fd e vlue of e RHS of (7) for e ( ) cosdero. I e bove emple s e ew mmum vlue of e RHS fuco. Oce s s ow oe hs o reur o e orgl problem d pply o ech erm e relo: ;. (5) Usg relos (5) e vlues of e vrbles re deermed for e model gve by relo (9). For e bove emple ese ew vlues re gve by o fve sgfc decml plces. 8

13 hp://sje.jourls.c.z Emple The purpose of s emple s o llusre vrous desg cosderos before ccepg prculr desg. osder coer s o be desged o hold cubc cm of lqud. Ths emple s slgh modfco of emple dscussed [] where oly e cyldrcl desg ws cosdered. Sce lrge umber of us of s produc would be requred e mmum cos per u s eremely mpor for e orgzo. Le e merl cos for e wlls be $. per sq cm wheres e cos of merl used for e boom d op of s coer s $.8 per sq cm. The cos of mufcurg coers s depede of e shpe of e coer. se osder crculr cyldrcl ube of rdus r d hegh h. The cos fuco o be mmsed s gve [] s gve by: T cylder ( ) ( ) () r.8π ( r ) r r (6) where d.8π. The opmly codos re gve by δ δ δ δ (7) The soluo for e bove equos s δ / δ /.e. e opml cos s dsrbued such / of e ol cos s owrds e wlls d / s for e op (/).8π (/) d boom. Ths resuls ol cos [ ( ).( ) ] 6.5. / / se osder coer w rgle ech sde of leg equl o d hegh h. 7 ( TΔ bse ) (.)( ) Δ Δ. (8) Noe epressos (6) d (8) re smlr hece opmly codos do o chge d e soluo rems e sme.e. δ / δ /. Thus for e 7 (/). (/) rgulr bse desg e cos would be: ( ).( ) 7.98 / / se oer w squre bse of sde d hegh h. The cos fuco s gve by: 9

14 hp://sje.jourls.c.z T squrebse.6 squre squre. (9) Oce g e cos equos for dffere desgs re e sme so e soluo rem uchged.e. δ / δ /. Thus e opml cos for s desg would be: ( / ).8 (/ ) ( ( / ) (/ ) ).( ) Thus mog ese ree desgs crculr bse s e les epesve. Oer desgs c be evlued smlr wy. Noe for e sme wegh f oe hs o cree lrge vsul mpresso rgle bse would be preferred. Recosder e bove desg problem d ssume grms of merl s o be pced oe of e shpes dscussed bove.e. crculr rgulr or squre bse. Assume desy s per cm. Relev formo hs bee summrsed e Tble below. Shpe of pce Fuco o be mmsed Surfce Are rculr bse ube r πr 9.65 Trgulr bse ube Squre Bse ube / r / / / π / / / r / / / / /.5 r r 9.66 / / Tble 7: Surfce res for uform quy ree dffere shpes Thus e rgulr shpe hs e mmum surfce re d my be e reso e vrous-szed chocoles w e brd me Tobleroe re mufcured rgulr shpes for beer vsul effec. 5. DISUSSION AND REMARKS. The suggesed pproch uses dchoomy o rele prese vlues o e mmede ps vlues.. By usg e pproch dscussed s pper my be cocluded opml

15 hp://sje.jourls.c.z vlues of o ll GPs re coeced by pece-wse ler p.. Ths soluo pproch my be desrble for problem epermel sge whe oe s cocered bou vrous vlues of desg prmeers before mplemeg fl desg.. If chges re relvely fewer s pproch my resul compuol svgs. See Emple dscussed seco where opmly codos for e GP models for e ree desgs dd o chge. Thus for ech of e subseque cosderos e prevous soluo remed vld. However o defe geerl seme c be mde s sge. Furer esg d supporg sofwre s ecessry before mg defe cocluso. 5. The de of formo recyclg meods c be rced my esg meods descrbed e lerure for emple e me seres lyss [89] recurrece equos used dymc progrmmg d reled corol processes [ ]. Furer my erler pproches dscussed [58] my lso be clssfed s formo recyclg meods. We beleve s de s lely o geere more eres furer reserch o me e memcl meods more megful for dusrl pplcos. 6. For furer reserch we hve o loo g s my memcl meods s possble d develop ppropre meods o cosder er lyss e proe suo rsg rel lfe pplcos. I my opo e feld s ope for reserch d your mgo s e lm. For o-ler cses e chllege s much more comple s o-ler fucos re much more sesve o chges compred w ler fucos. 7. Problem-solvg s le wg wr. Ths logy ws esblshed [58]. I s pper e m s o emphsze model s represeo of busess suo whch chges frequely le wr suo. Therefore o be ble o del w relsc suo model mus be ble o bsorb shocs rse due o uepeced ercos. How bumpy busess rde c be s clerly refleced e followg wo quoes from rece boo Srgh from e gu by Jc Wlsh brll busess mgc [8]:. Pge XV prologue: There s o perfec busess sory. I beleve busess s lo le world-clss resur. Whe you pee behd e che doors e food ever loos s good s whe comes o your ble o fe ch perfecly grshed. Busess s messy d choc.. Pge 77: Beg EO s e us! A whole jumble of oughs come o md: Over e op. Wld. Fu. Ourgeous. rzy. Psso. Perpeul moo. The gve-d-e. Bg decsos e rel gme. rses d pressure. Los of swgs. A few home rus. The rll of wg. The p of losg.

16 hp://sje.jourls.c.z Noe models re o help ese busess eecuves who fd emselves lwys o roller-coser rde. If e model s eded o help em hs go o be ble o rde o e sme roller-coser. Ths pper emps lyss of geomerc progrm chgg evrome. 6. AKNOWLEDGEMENTS Ths re due o oymous referees for er cosrucve suggesos. 7. REFERENES [].S. Beghler d D.T. Phlps Appled Geomerc Progrmmg Joh Wley & Sos 976. [] R. Bellm Dymc progrmmg Prceo Uversy Press 957. [] R. Bellm d S. Dreyfus Appled Dymc Progrmmg Prceo Uversy Press 96. [] S.. Fg d S. Puepur Ler Opmso d Eesos Theory d Algorms Prece Hll 99. [5] W.A Fuller Iroduco o Sscl Tme Seres Joh Wley d Sos 996. [6].B. Grc d W.I. Zgwll Pwys o Soluos Fed Pos d Equlbr Prece Hll Seres ompuol Memcs 98. [7] V.I. Grover V.K. Grover d W. Hogld Recoverg Eergy from Wse: Vrous Aspecs Scece Publshers Efeld (NH) USA. [8] J.D. Hmlo Tme Seres Alyss Prceo Uversy Press 99. [9] M.S. Klm Memcl Modellg lssroom Noes Appled Memcs SIAM 987. [] V. Klee d G.L. My How good s e Smple Meod? Iequles III Ed. O.Shsh Acdemc Press New Yor [] S. Kumr Opmso of Proe Sysems A Revew APORS 99 (Eds.) M. Fushm d K. Toe World Scefc Publshers pp [] S. Kumr H. Aror. her d G. Rvgesh Ps for self-helg Proe Newor Pdm Bhus Professor B.N. Prsd Br eery ommemorve volume publshed by e Allhbd Memcl Socey Bulle of Allhbd Memcl Socey Vol. pp [] S. Kumr G. Rvgesh J. McPhee d T.M. Ny Terml Pr relbly Proe Newor Ierol Jourl of Operos d Quve mgeme Vol. 5 No. pp [] S. Kumr Iformo Recyclg Memcl meods: A Phlosophy d hllege hper Imporce of Memcl Modellg of Bologcl d Bomedcl Processes Eds. LS Luboob JYT Mugsh d J. Ksoz Afrc Socey of Bomemcs Seres No. Merere Uversy Press Kmpl Ugd pp -6. [5] S. Kumr Iformo Recyclg Memcl meods for Proe Sysems: A P-wy Approch Sou Afrc Jourl of Idusrl Egeerg Vol. 6 No. pp 8-5. [6] HM Tluder d S. Kumr rcus free of specfed ls proe dsrbuo ewor Dscovery d Iovo Vol. 7 Nos. d pp7-

17 hp://sje.jourls.c.z 5. [7] HM Tluder S. Kumr d B Joes Iformo Recyclg Memcl Meods: Alyss of l weghed proe commuco flow ewors o pper Id Jourl of Memcs Vol. 8 No. 6.. [8] Su Tzu The r of wr Shmbhl Drgo Edos 988. [9] J. Welsh Srgh from e gu Wrer Busess boos.