bsc thesis - an enhanced communication discrete particle swarm optimization algorithm for the...

7/25/2019 BSc Thesis - An Enhanced Communication Discrete Particle Swarm Optimization Algorithm for the Travelling Sales

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cfrfmy lfahjrf tcjt tcbs prodfat work wjs ajrrbfl out my ef jgl fvfry otcfr fxtfrgjh work

usfl bg tcbs prodfat work cjs mffg luhy jakgowhflifl.

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Vgbvfrsbty o` Dos, ^hjtfju Qtjtf, Gbifrbj.

___________________________ __________________

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(Qupfrvbsor)

___________________________ _____________________

Lr. \jgko Bscjyj Ljtf

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5/128

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tcrouic tcbs uglfrirjlujtf lfirff. O` j trutc, cf cjs mffg tcf Ogf wco cjs pfr`fatfl jhh

tcbgis òr ef. B je gotcbgi wbtcout Cbe. Ybtc jhh ey cfjrt, B sjy, \cjgk you @jtcfr.

Ey sbgafrf jpprfabjtbog iofs to ey pjrfgts, ^jstor & Ers. ôpoohj, Q. @. òr tcfbr

hovf, ajrf, fgaourjifefgt, jgl bgvjhujmhf jssbstjgaf tcrouicout tcf pfrbol o` tcbs proirje. B

lffphy rfaoigbzf jgl jpprfabjtf tcf fxafptbogjh sjarb`bafs tcfy cjvf ejlf to ejkf got dust

tcbs lfirff j suaafss, mut bglffl ey wcohf hb`f. Rou jrf tcf mfst pjrfgts bg tcf worhl, jgl B

tcjgk Iol òr tcf prbvbhfif o` aoebgi bgto tcbs worhl tcrouic you. B jhso wjgt to jpprfabjtf

ey sbmhbgis, ^fjaf, ^rjbsf jgl ^juh òr tcfbr aogstjgt support jgl fgaourjifefgt. Rou cjvf

mffg ey `rbfgls jgl pjrtgfrs bg proirfss, jgl j mhfssbgi to ef. B tcjgk Iol òr you, jgl ejy

Iol hb`t you up.

B ajggot mut scow ey lffp jpprfabjtbog to tcf mfst supfrvbsor bg tcf worhl, Lr. G.P.

Mhjejc. B hjak worls to jpprfabjtf jhh tcf hovf, ajrf jgl ugaoeeog aogafrg you cjvf scowg

towjrls ef. Rou ejlf wrbtbgi tcbs prodfat j phfjsurf jgl jg jlvfgturf, wbtc jhh your jlvbaf,

fgaourjifefgt jgl acjhhfgifs to ef. B rfjhhy logt kgow wcjt to sjy jmout jhh tcf sjarb`bafsyou ejlf òr ef, cfhpbgi ef bg so ejgy wjys wcbac B ajggot sjy cfrf. B mfhbfvf wbtc jhh ey

cfjrt tcjt our efftbgi jgl workbgi toiftcfr bs Iol-orljbgfl, jgl tcjt wf wbhh aogtbguf to

work toiftcfr mfyogl tcbs hfvfh. \cjgk you so euac, sbr.

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jrf j Iol-sfgl. Rou cfhpfl ef so euac, out o` ejgy muiiy sbtujtbogs, jgl wf cjl so euac

ùg bg tcf proafss. \f iusto euaco, eb jebio ejrjvbhhoso (jgl B efjg tcjt bg j iool sfgsf!

;) ).

Ejgy tcjgks to ey irfjt `rbfgls o` 1YA `jef, Mbtrus Yugb jgl Jlftjyo Qyhvfstfr.

Rou jrf tcf mfst. Yfvf mffg toiftcfr rbict `roe wcfg wf wfrf irffgcorgs bg :>>-hfvfh tbhh

gow, jgl wfvf cjl ejgy fxpfrbfgafs toiftcfr. Mfbgi your `rbfgl cjs mffg j mhfssbgi.

Abg`wjt Loijk, jkj ^rombty, you cjvf jhso mffg j vfry iool `rbfgl. \cjgks òr jhh tcf blfjs

wfvf scjrfl jgl jhh tcf ùg wfvf cjl. Bts jhwjys j phfjsurf to tjhk wbtc you. Got to òrift

ey irfjt `rbfgls bg VDÂ, Ejmjs, Dulf, Feejgufh, Omb, Fglff, Fefkj, Gjtcjgbfh, Gjsbru,


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phjafs.

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Glukwf jgl so ejgy otcfrs or tcfbr bepjat og ef lurbgi ey stjy bg tcf Vgbvfrsbty o` Dos.

B ajggot lo wbtcout jakgowhflibgi tcf bepjat o` tcf LHA@, Vgbdos og ey hb`f jhh

tcrouic tcf woglfrùh tbef B cjvf spfgt wbtc tcfe. Yortcy o` spfabjh efgtbog jrf ey vbhhjif

costfh mrotcfrs jgl sbstfrs. Rouvf jhh mffg j irfjt jllbtbog jgl mhfssbgi to ey hb`f. B wjgt to

spfabjhhy jpprfabjtf ey woglfrùh pjstor, ^jstor Ohuimfgij Oiugòwokjg. Rour bepjat bg

ey hb`f wbhh gfvfr mf òriottfg.

@bgjhhy, ey jpprfabjtbog iofs to jhh ey `rbfgls, aoursf ejtfs jgl wfhh-wbscfrs. Ejytcf iool Horl bg cbs bg`bgbtf efrabfs mhfss you jhh (Jefg).


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J^^]OPJH ........................................................................................................................... bbb

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JAKGOYHFLIFEFG\Q .................................................................................................... v

\JMHF O@ AOG\FG\Q ...................................................................................................... vbb

JMQ\]JA\ .............................................................................................................................. x

ACJ^\F] OGF4 BG\]OLVA\BOG .................................................................................... :

:.: Mjakirougl ....................................................................................................... :

:.:.: Mrbf` Cbstory o` \Q^ .................................................................................. :

:.:.= Mrbf` Cbstory o` Jhiorbtces ....................................................................... :

:.:.1 Mrbf` Cbstory o` Qwjre Bgtfhhbifgaf .......................................................... =

:.:.


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9/128

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]f`frfgafs ............................................................................................................................... 93

Jppfglbx B .............................................................................................................................. 36

Jppfglbx BB ............................................................................................................................. 39


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JMQ\]JA\

\cf syeeftrbajh \rjvfhhbgi Qjhfsejg ^romhfe (\Q^) bs stjtfl tcus4 j trjvfhhbgi sjhfsejg cjsto vbsbt fjac ogf o` j hbst o` x abtbfs fxjathy ogaf jgl rfturg to tcf abty wcfrf cf stjrtfl `roe.\cf aost o` trjvfhhbgi mftwffg fjac sft o` abtbfs bs kgowg, jgl bs tcf sjef, brrfspfatbvf o`lbrfatbog. Ycbac o` tcf possbmhf tours ajg cf tjkf jt ebgbeue totjh aost? Pjrbous jhiorbtcescjvf mffg lfvfhopfl to jpproxbejtf sohutbogs to tcbs ahjss o` promhfes, jgl ogf ahjss o` suacjhiorbtces bs Lbsarftf ^jrtbahf Qwjre Optbebzjtbog. \cbs prodfat fxphjbgs vjrbous sjhbfgtaogafpts bg gp-aoephftfgfss tcfory jgl eftjcfurbstba rfsfjrac, jgl proposfs j gfw cfurbstba,Fgcjgafl Aoeeugbajtbog Lbsarftf ^jrtbahf Qwjre Optbebzjtbog (FA-L^QO), wcbaclbsarftbzfs tcf ^QO eftjcfurbstba, js wfhh js bgtroluafs two kfy aogstjgts, K jgl \ wcbac jrfusfl bg bephfefgtbgi j govfh aogafpt o` bg`hufgabgi tcf proafss tcrouic wcbac tcf sfhfatbogo` flifs bg tcf sfjrac proafss bs ejlf, my lbrfatbgi tcf proafss towjrls eorf proebsbgiflifs jgl jwjy `roe hfss proebsbgi ogfs. \cbs cjs mffg logf wbtcout sjarb`babgi tcffxphorbgi jgl fxphobtbgi propfrtbfs o` tcf ^QO. ]jtcfr, tcfy cjvf mffg beprovfl. \cf

jhiorbtce bs tfstfl og < to =>-abty bgstjgafs o` \Q^. ]fsuhts scow tcjt tcf bgtroluatbog o` tcfaogstjgts K jgl \ irfjthy beprovfs tcf spffl o` aogvfrifgaf jgl jaaurjay o` tcf L^QO, jhsoirfjthy rfluabgi bts tfglfgay to ift stuak og hoajh ebgbej. Jhso, j proirje wjs lfvfhopfl tofxfautf tcf jhiorbtce jgl scow j vbsujh rfprfsfgtjtbog o` tcf bgput jgl output ljtj, ibvbgirfsfjracfrs j usf`uh jgl vbsujhhy jppfjhbgi so`twjrf `or sbeuhjtbgi tcf jhiorbtce.


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ACJ^\F] OGF4 BG\]OLVA\BOG

:.: Mjakirougl

Ejg js j soabjh fgtbty cjs jhwjys mffg aogafrgfl wbtc lbsaovfrbgi cbs promhfes, stulybgi

tcfbr gjturf, jgl `bglbgi efjgs o` sohvbgi tcfe, or `bglbgi efjgs o` beprovbgi fxbstbgi

systfes so js to jacbfvf irfjtfr fjsf jgl aoeòrt. Ybtc tcf pjssjif o` tbef, ejg cjs mffg

jmhf to, bg tcf aoursf o` jttfeptbgi promhfe sohvbgi, fstjmhbsc soef systfejtba jpprojacfs to

rfsfjracbgi, lbsaovfrbgi tcf gjturf o`, jgl sohvbgi cbs promhfes. Bg tcf ebllhf :21>s

aoeputfr sabfgaf wjs yft j got wfhh-lf`bgfl jajlfeba lbsabphbgf. Jatujhhy, ùgljefgtjh

aogafpts, suac js jhiorbtce, or aoeputjtbogjh promhfe, cjl mffg òrejhbzfl dust soef

yfjrs mfòrf (Irfao @. , =>>3).

:.:.: Mrbf` Cbstory o` \Q^

\cf \rjvfhhbgi Qjhfsejg ^romhfe (\Q^) bs pjrt o` tcf ahjss o` gog-lftfrebgjtf pohygoebjh-

tbef cjrl (gp-aoephftf) promhfes, wcbac jrf promhfes tcjt ajggot mf fxjathy sohvfl. \cjt bs,

tcfrf bs go jgjhytbajh wjy o` proluabgi sohutbogs to gp-aoephftf promhfes. Ejtcfejtbajh

promhfes rfhjtfl to tcf \Q^ wfrf `brst stulbfl bg tcf :3>>s my tcf ejtcfejtbabjgs Qbr

Ybhhbje ]owjg Cjebhtog jgl \coejs ^fgygitog Kbrkejg. \cf ifgfrjh òre o` tcf \Q^

aouhl cjvf mffg `brst stulbfl bg :21> my Kjrh Efgifr (Aook, =>>9). Eftcols òr sohvbgi tcf\Q^ stjrtfl jppfjrbgi bg pjpfrs bg tcf ebl-:26>s, jgl tcf promhfe cjs mffg sumsfqufgthy

stulbfl my ejgy rfsfjracfrs rjgibgi `roe ejtcfejtbas, aoeputfr sabfgaf, acfebstry,

pcysbas, jgl otcfr sabfgafs.

:.:.= Mrbf` Cbstory o` Jhiorbtces

Jg jhiorbtce aouhl mf lf`bgfl js j aogabsf jgl `bgbtf sft o` proaflurfs or stfps lfsarbmfl òr

aoephftbgi j tjsk or sohvbgi j promhfe. \cf tfre jhiorbtce wjs aobgfl js j trbmutf to Jmu

Jmljhhjc Eucjeejl bmg Eusj Jh-Kcwjrbzeb, j ^frsbjg ejtcfejtbabjg wco hbvfl Mjicljl

jrougl 3>>MA (Frbaksog, =>:1), wco wrotf tcf fjrhbfst kgowg ejtfrbjh aogtjbgbgi j prfabsf

proaflurf òr sohvbgi jhifmrjba promhfes suac js jllbtbog, euhtbphbajtbog, lbvbsbog, squjrf

roots jgl pb ajhauhjtbog, jgl bs mfhbfvfl to mf tcf `brst to usf tcf `biurf > js j phjaf cohlfr bg

posbtbogjh mjsf gotjtbog (Hbmfrjtorf, :223). Qbgaf tcf òrejh bgtroluatbog o` tcf tfre, ejgy

lb``frfgt `hjvours o` jhiorbtces jgl jpprojacfs to promhfe-sohvbgi cjvf fvohvfl.


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:.:.1

Mrbf` Cbstory o` Qwjre Bgtfhhbifgaf

Qwjre Bgtfhhbifgaf bs j `bfhl o` aoeputfr sabfgaf tcjt lfsbigs jgl stulbfs f``babfgt

aoeputjtbogjh eftcols òr jpproxbejtbgi sohutbogs to gp-aoephftf promhfes bg j wjy tcjt bs

bgspbrfl my tcf mfcjvbour o` rfjh swjres or bgsfat aohogbfs (Efrkhf & Ebllfglor`, =>>6).\cf `jsabgjtbgi jmbhbty o` ugsopcbstbajtfl bglbvblujhs bg tcfsf rfjh swjres or aohogbfs to (my

ajrrybgi out jatbogs jgl bgtfrjatbogs o` surprbsbgi sbephbabty) sohvf aoephfx promhfes jgl

fxcbmbt aoephfx orijgbzjtbog jgl aoorlbgjtbog wbtcout afgtrjh aogtroh o` jgy kbgl (wcbac

jmbhbty bs kgowg js fefrifgaf) bs wcjt cjs jttrjatfl motc mbohoibsts jgl aoeputfr sabfgtbsts

to motc stuly tcfe, jgl jpphy tcfbr eftcols bg sohvbgi hb`fs promhfes jgl acjhhfgifs. \cf

orbibg o` swjre bgtfhhbifgaf aouhl mf trjafl so `jr js to tcf stuly o` tcf shbef eohl jrougl

:203 my j eohfauhjr mbohoibst jgl pcysbabst gjefl Fvfhyg @ox Kfhhfr, jgl jg jpphbflejtcfejtbabjg gjefl Hff Qfifh. \oiftcfr tcfy wfrf jmhf to, bg j rfvohutbogjry lbsaovfry,

fxphjbg tcf so-ajhhfl lbsjppfjrbgi jgl sumsfqufgt rf-jppfjrbgi jat o` tcf shbef eohl mjsfl

og acjgifs bg tcf fgvbrogefgt. \cf popuhjr mfhbf` bg tcosf ljys wjs tcjt tcf lfabsbog o` tcf

fgtbrf shbef eohl, jgl eorf mjsbajhhy, tcf bglbvblujh afhhs tcjt òrefl tcf shbef eohl to

fbtcfr jiirfijtf or lbsbgtfirjtf wjs aogtrohhfl my j `fw pjaf-ejkfr afhhs (afhhs wcbac go

ogf wjs jmhf to blfgtb`y). Cowfvfr, aogtrjry to tcjt mfhbf`, Kfhhfr jgl Qfifh provfl tcjt tcf

lfabsbog o` tcf fgtbrf eohl wjs j rfsuht o` jg jiirfijtbog o` tcf bglbvblujh `bglbgis jgljatbogs o` tcf aogstbtufgt afhhs, wbtcout jgy afgtrjh aogtroh (Docgsog, =>>:). \cf tfre

Qwjre Bgtfhhbifgaf wjs `brst bgtroluafl my I. Mfgb jgl D. Yjgi bg :232 js j sft o`

jhiorbtces òr aogtrohhbgi romotba swjres (Jcefl & Ihjsiow, =>:=).

:.:.:>; Kfggfly & Fmfrcfrt, :226). \cf jhiorbtce cjs lrjwg tcf

bgtfrfst o` ejgy sabfgtbsts jaross lb``frfgt `bfhls mfajusf o` bts sbephbabty o` bephfefgtjtbog,

bts tfglfgay to qubakhy aogvfrif to j rfjsogjmhy iool sohutbog, jgl bts jmbhbty to lfjh wbtc

lb``bauht optbebzjtbog promhfes f``babfgthy, js ajg mf sffg my tcf vjst vohuef o` rfsfjrac

work tcjt cjs mffg lflbajtfl to bt wbtcbg tcf scort tbef bg wcbac bt cjs mffg lfvfhopfl. Bg

`jat, js jt =>:>, ovfr :6>> lb``frfgt pjpfrs cjvf mffg pumhbscfl og ^QO jhogf (^jrsopouhos


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1

& Prjcjtbs, =>:>; Kcjgfsjr ft jh, =>>2; Ajb ft jh, =>>2). \cf ^jrtbahf Qwjre Optbebzjtbog

jhiorbtce ifgfrjhhy cjs j popuhjtbog o` pjrtbahfs (ajhhfl tcf swjre), j sfjrac spjaf o`

possbmhf sohutbogs, jgl j `btgfss ùgatbog wcbac lftfrebgfs tcf lfsbrjmbhbty o` j pjrtbauhjr

sohutbog.

:.= Jbe jgl Omdfatbvfs

\cbs prodfat jbes to òreuhjtf jg jhiorbtce òr sohvbgi tcf \Q^, jgl lfvfhop jg jpphbajtbog

wcbac wbhh mf jmhf to jpproxbejtf j sohutbog to jgy promhfe wcbac ajg mf eolfhhfl js j

\rjvfhhbgi Qjhfsejg ^romhfe, usbgi tcf ^jrtbahf Qwjep Optbebzjtbog tfacgbquf. Bg orlfr to

jacbfvf tcbs, tcf rfsfjracfr cjs aoef up wbtc tcf òhhowbgi omdfatbvfs4

:. Ibvf j prjatbajh jgl lowg-to-fjrtc fxphjgjtbog o` tcf \rjvfhhbgi Qjhfsejg ^romhfe

jgl mrbf`hy lbsauss G^-aoephftfgfss tcfory.

=. Fxjebgf tcf `bfhl o` Qwjre Bgtfhhbifgaf jgl fxphjbg j `fw Qwjre Bgtfhhbifgaf

jhiorbtces, wbtc prbejry òaus og ^jrtbahf Qwjep Optbebzjtbog (cfgafòrtc ^QO)

1. Vsf tcf ahjssbajh \rjvfhhbgi Qjhfsejg ^romhfe (cfgafòrtc \Q^) js j ajsf stuly so js

to lfeogstrjtf tcf prjatbajmbhbty o` ^QO bg pro``frbgi jpproxbejtf sohutbogs to

promhfes eolfhhfl js syeeftrba \Q^s.

>3), jgl ajg mf sueejrbhy lfsarbmfl jsòhhows4 j sjhfsejg cjs to vbsbt g abtbfs, fjac fxjathy ogaf, stjrtbgi `roe jgl fglbgi jt j

pjrtbauhjr lfsbigjtfl abty. Kgowbgi tcf aost o` trjvfhhbgi mftwffg fjac abty, wcjt bs tcf mfst

orlfr wbtc wcbac cf ajg vbsbt jhh tcf rfqubrfl abtbfs suac tcjt tcf ovfrjhh aost bs ebgbebzfl? Bg

ifgfrjh, tcbs stuly jbes to pro``fr j efjgs o` sohvbgi jgy promhfe bg wcbac jg optbejh

orlfrbgi o` j guemfr o` btfes cjs to mf acosfg bg suac j wjy tcjt tcf totjh aost o` j sohutbog

bs lftfrebgfl my jllbgi up tcf aosts jrbsbgi `roe iobgi `roe ogf btfe to j suaafssbvf ogf. Bg

tcf syeeftrba ajsf wcbac tcf stuly bs to mf gjrrowfl to, tcf aost o` trjvfhhbgi `roe abty b todbs fqujh to tcf aost o` trjvfhhbgi `roe abtyd to abty b.\cjt bs, acjgif bg tcf lbrfatbog o` `how


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> prbzf. \cf promhfe wjs stjtfl tcus4

Bejibgf tcjt \ooly jgl Euhloog wjgt to lrbvf jrougl tcf aougtry jgl vbsbt fjaco` tcf 11 hoajtbogs rfprfsfgtfl my lots og tcf aogtfst ejp, jgl tcjt bg lobgi so,

tcfy wjgt to trjvfh tcf scortfst possbmhf routf. Rou scouhl phjg j routf `or tcfe


16/128

0

`roe hoajtbog to hoajtbog wcbac wbhh rfsuht bg tcf scortfst totjh ebhfjif `roeAcbajio, Bhhbgobs mjak to Acbajio, Bhhbgobs.

Eorf ifgfrjhhy, you jrf ibvfg j aohhfatbog o` abtbfs jgl tcf lbstjgaf to trjvfh mftwffg tcfe.

Rou jrf to `bgl tcf scortfst routf to vbsbt fjac abty jgl to rfturg to tcf stjrtbgi pobgt. \cbs

sougls fjsy `roe ogf pobgt o` vbfw. Jhh wf cjvf to lo bs fguefrjtf jhh tcf possbmhf routfs

jgl `bgl tcf scortfst, sbgaf tcfrf jrf oghy j `bgbtf guemfr o` possbmhf routfs tcrouic j ibvfg

sft o` abtbfs. Cowfvfr, bg tcbs pjrtbauhjr promhfe, you wouhl gffl to fxjebgf

:1:,606,>0,>3>,>>>,>>> lbstbgat tours. Fr,you ejy mfibg to

sjy, sarjtacbgi your cfjl, jcj, wf aouhl fjsbhy lo tcjt wbtc tcf fxtrfefhy `jst aoeputfrs wf

cjvf toljy! Got so fjsy.\cf BME ]ojlruggfr Ahustfr o` tcf Vgbtfl Qtjtfs Lfpjrtefgt o`

Fgfriy cjs :=2,0>> aorfs jgl toppfl tcf =>>2 rjgkbgi o` tcf 6>> worhls `jstfst

supfraoeputfrs, fxfautbgi up to :, >0).

=.:.:

\rjatjmhf jgl Bgtrjatjmhf ^romhfes

Yf mfibg wbtc j sbephbstba lf`bgbtbog o` j promhfe4 J qufstbog proposfl `or sohutbog

(Yfmstfr). Aogsblfr tcf promhfe o` euhtbphybgi two bgtfifrs usbgi oghy jllbtbog or

sumtrjatbog opfrjtbogs usbgi j aoeputfr. @bglbgi j sohutbog to tcf promhfe o` euhtbphybgi tcf

bgtfifrs =1 jgl


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9

\rjatjmhf ^romhfes

Lf`bgbgi tcf tfres trjatjmhf jgl bgtrjatjmhf wbtc tcf jmovf lf`bgbtbog o` tcf tfre sohutbog bg

ebgl, j trjatjmhf promhfe bs j promhfe wcbac jlebts jg jhiorbtce wbtc worst-ajsf tbef-

lfejgl bg O(ga) òr soef bgtfifr a (Iorsf, =>::). Bg otcfr worls, j trjatjmhf promhfe bs ogfwcbac ajg mf sohvfl bg pohygoebjh tbef. \cbs lofs got efjg tcjt jhiorbtces wcbac ajggot

sohvf suac j promhfe bg pohygoebjh tbef lo got fxbst. Bt efjgs, rjtcfr, tcjt ogaf j sbgihf

jhiorbtce fxbsts, wcbac ajg sohvf j promhfe wbtc worst-ajsf tbef-lfejgl mouglfl my O(ga)

(a mfbgi bg tcf rjgif >, :, ), tcf promhfe bs trjatjmhf. Jhh trjatjmhf promhfes mfhogi to tcf

ahjss o` promhfes lfgotfl my ^. Fxjephfs o` trjatjmhf promhfes bgahulf ejtrbx

euhtbphbajtbog, gftwork `how, sbgihf-souraf scortfst pjtc, sfjracbgi, sortbgi jgl ebgbeue

spjggbgi trff promhfes (Mrblif, =>>6j). \cbs rfsfjrac lofs got hjy js euac fepcjsbs ogtrjatjmhf promhfes, mut òausfs og bgtrjatjmhf ogfs.

Bgtrjatjmhf ^romhfes

\cfsf jrf promhfes òr wcbac go sohvbgi jhiorbtce bs kgowg. Bt cjs mffg scowg tcjt òr tcfsf

promhfes, tcf howfr mougl og tcf tbef-lfejgl o` jgy possbmhf jhiorbtce to sohvf tcfg bs j

ùgatbog tcjt irows ugrfjsogjmhy `jst (Iorsf, =>::). \cbs acjrjatfrbstba o` bgtrjatjmhf

promhfes bs jhso kgowg js tcf aursf o` lbefgsbogjhbty.\cbs, cowfvfr, lofs got efjg tcjt

go bgstjgaf o` suac promhfes cjs mffg sohvfl. Ifgfrjhhy, bgstjgafs o` bgtrjatjmhf promhfes

wbtc rfhjtbvfhy sejhh bgput sbzfs ajg mf sohvfl bg rfjsogjmhf tbef (Hbmfrjtorf, :223). Yf

òrejhhy lf`bgf bgtrjatjmhf promhfes tcfrfòrf js promhfes tcjt ajggot mf sohvfl my j

pohygoebjh-tbef jhiorbtce, tcjt bs, tcf howfr mougl o` tcfbr tbef-aoephfxbty bs fxpogfgtbjh

(Mrblif, =>>6j).

=.=

G^-Aoephftfgfss

G^-Aoephftfgfss tcfory jrbsfs `roe promhfes wcbac, js fjrhbfr stjtfl, cjvf tcf propfrty o`tcf aursf o` lbefgsbogjhbty. \cfy jrf j ahjss o` promhfes wcbac cjvf lf`bfl f``babfgt

(pohygoebjh-tbef) jhiorbtces, lfspbtf lfajlfs o` bgtfgsf rfsfjrac (Iopjhjkrbscgjg, =>>0).

\cfsf promhfes aouhl mf sohvfl, bg tcf blfjh sfgsf, my fguefrjtbgi jhh possbmhf sohutbogs jgl

acfakbgi tcfe ogf my ogf to sff wcbac ogf bs tcf aorrfat ogf. Cowfvfr, js cjs mffg

bhhustrjtfl jt tcf mfibggbgi o` tcf acjptfr, suac jg jhiorbtce, wcosf ruggbgi tbef wbhh

lf`bgbtfhy mf supfr-pohygoebjh, bs jhh mut usfhfss bg prjatbaf. \cfrfòrf, tcf jbe o` rfsfjracfrs

cjs mffg to mypjss fxcjustbvf sfjrac jgl, usbgi ahufs `roe tcf bgput, irfjthy gjrrow lowg


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3

tcf sfjrac spjaf to jg fxtfgt wcfrf j aorrfat sohutbog ajg mf òugl bg pohygoebjh tbef

(Ljsiuptj, ^jpjlbebtrbou, & Pjzbrjgb, =>>0).

=.=.: Lftfrebgbse jgl Gog-Lftfrebgbse

\o lf`bgf tcf tfres gp-aoephftf jgl gp-cjrl wf `brst gffl to uglfrstjgl tcf tfres

lftfrebgbse jgl gog-lftfrebgbse. @roe rfvbfwfl hbtfrjturf, gp-cjrlgfss bs lf`bgfl bg tfres

o` Lftfrebgbstba \urbgi Ejacbgfs (L\E) jgl Gog-Lftfrebgbstba \urbgi Ejacbgfs (GL\E).

J \urbgi Ejacbgf bs ogf wcbac cjs two lb``frfgt cjhtbgi stjtfs, ajhhfl tcf jaafptbgi jgl

rfdfatbgi stjtfs. J \urbgi ejacbgf, E bs sjbl to mf mouglfl my tbef \, b` òr jgy bgput x, E

og x jhwjys cjhts bg \(|x|) eovfs/stfps (Yjgi, =>>0). Qbephy put, jg jhiorbtce bs j \urbgi

Ejacbgf (Eourj, =>>=), tcouic bg tcbs ajsf wf ejy rfstrbat bt to jg jhiorbtce òr sohvbgi

lfabsbog promhfes. Cfgaf, j \urbgi ejacbgf E bs lftfrebgbstba b`, ibvfg tcf stjtf o` tcf

ejacbgf jgl tcf stjtf o` tcf efeory, tcf gfxt eovf to mf ejlf bs ugbqufhy lftfrebgfl. Og

tcf otcfr cjgl, E bs gog-lftfrebgbstba b`, ibvfg tcf ejacbgf jgl efeory stjtfs, tcfrf jrf

eorf tcjg ogf possbmhf eovfs, `roe wcbac, fbtcfr ogf bs sfhfatfl tcrouic iufssbgi, or jhh tcf

possbmhf eovfs jrf fxfautfl jt tcf sjef tbef (Hbmfrjtorf, :223; Gbaosbj, =>>2). Bg tcf ajsf

wcfrf j sohutbog bs iottfg tcrouic iufssbgi, tcf rfsuht scouhl mf vfrb`bjmhf bg pohygoebjh

tbef.

=.=.=

Ahjss o` G^

\cf tfre G^ stjgls òr Gog-lftfrebgbstba ôhygoebjh tbef, jgl promhfes bg tcbs ahjss jrf

promhfes wcbac jrf sohvjmhf usbgi jg GL\E bg pohygoebjh tbef (Yjkbh, =>:1). \cf G^

ahjss ajg jhso mf lf`bgfl js tcf ahjss o` promhfes wcbac jrf vfrb`bjmhf bg pohygoebjh tbef,

tcf tfre vfrb`bjmhf efjgbgi tcjt b` j afrtb`bajtf o` j sohutbog wfrf soefcow to mf ibvfg,

tcfg tcf aorrfatgfss o` tcf afrtb`bajtf aouhl mf vfrb`bfl bg tbef pohygoebjh bg tcf sbzf o` tcf

bgput to tcf promhfe (Aorefg, Hfbsfrsog, ]bvfst, & Ahb`òrl, =>>2). Bg sbephfr tfres, j

promhfe bs bg tcf G^ ahjss b` j sohutbog to bt ajg mf vfrb`bfl to mf aorrfat wbtcbg pohygoebjh

tbef. Bt ajg mf omsfrvfl `roe tcf jmovf lf`bgbtbogs tcjt jhh promhfes mfhogibgi to ^ jhso

mfhogi to G^. \cf qufstbog o` wcftcfr promhfes mfhogibgi to G^ jhso mfhogi to ^ bs jg

bgtfrfstbgi ogf wcbac wbhh mf trfjtfl sumsfqufgthy.

ôhygoebjh-tbef ]fluatbog

Mfòrf j òrejh lf`bgbtbog o` tcf tfre gp-cjrlajg mf ibvfg, wf gffl to prfsfgt tcf efjgbgi

o` tcf tfre pohygoebjh-tbef rfluatbog bg gp-aoephftfgfss tcfory. Jaaorlbgi to Lrfwfs


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2

(=>:1), Hft J jgl M mf lfabsbog promhfes. J pohygoebjh-tbef rfluatbog `roe J to M bs j

ùgatbog`suac tcjt4

@or fvfry bgputx,xJ `(x) M jgl

` ajg mf aoeputfl bg (lftfrebgbstba) pohygoebjh tbef

Yf sjy tcjt J bs pohygoebjh-tbef rfluabmhf to M, jgl wrbtf J pM.

Iorsf (=>::) lf`bgfs tcf tfre pohygoebjh-tbef rfluatbog bg tcbs wjy4 \o sjy j promhfe J

rfluafs bg p-tbef to jgotcfr ogf, M, wrbttfg js J pM efjgs tcjt tcfrf bs soef proaflurf,

wcbac tjkfs jt eost pohygoebjh tbef js j ùgatbog o` tcf sbzf o` tcf bgput to J, jgl wcbac

aogvfrts jg bgput bgstjgaf o` J bgto jg bgput bgstjgaf o` M

jhhows j subtjmhf jhiorbtce òr promhfe M to mf fxfautfl

provblfs j efacjgbse wcfrfmy tcf output omtjbgfl my tcbs jhiorbtce òr promhfe M

ajg mf trjgshjtfl mjak bgto jg output òr promhfe J.

\cf jhiorbtce òr promhfe M cfgaf jhso provblfs j sohutbog to promhfe J. @urtcfreorf, tcf

jhiorbtce wcbac sohvfs J eust jhso mf bg tcf sjef aoephfxbty ahjss js tcjt wcbac sohvfs M.

^bvotjhhy, b` wf kgow or suspfat tcjt wf cjvf j howfr mougl og tcf tbef lfejgl o` jhh

possbmhf jhiorbtces òr M, wf ajg sjy tcjt, bg tfres o` bts ùgljefgtjh lb``bauhty, promhfe J

bs go worsf tcjg promhfe M.

=.=.1 G^-Cjrl jgl G^-Aoephftf ^romhfes

Cjvbgi lf`bgfl jhh tcf rfqubrfl tfres, wf prfsfgt tcf lf`bgbtbog o` tcf tfres G^-cjrl jgl

G^-aoephftf4 B` òr fjac promhfe J G^, J pM, tcfg M bs G^-cjrl. M bs G^-aoephftf b` bt

bs j efemfr o` G^ jgl bs G^-cjrl (Hbmfrjtorf, :223). Mfajusf o` tcf rfluatbog rfqubrfefgt

òr gp-aoephftf promhfes, promhfes bg tcbs ahjss cjvf tcf pfauhbjr acjrjatfrbstba tcjt b` jgy

ogf o` tcfe ajg mf sohvfl bg pohygoebjh tbef, tcfg jhh o` tcfe ajg mf sohvfl bg pohygoebjh

tbef. G^-aoephftf promhfes jrf tcf cjrlfst bg G^.

J lbsaussbog og gp-aoephftfgfss wouhl sffe to mf bgaoephftf wbtcout tcf efgtbog o` tcf

sjtbs`bjmbhbty promhfe (QJ\), wcbac, jaaorlbgi to Hbmfrjtorf (:223) bs tcf prototypbajh G^-

aoephftf promhfe. QJ\ wjs provfl to mf gp-aoephftf my Qtfvf Aook bg :29:, jg jssfrtbog

wcbac bs gow kgowg js Aooks \cforfe(Iorsf, =>::). Iolljrl (=>>


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QJ\ bs j promhfe wcbac, ibvfg j moohfjg òreuhj bg AG@ jsks b` tcfrf bs j possbmhf

jssbigefgt wcbac ejkfs tcf moohfjg òreuhj truf. \cjt bs, tcf bgput to j QJ\ bs j moohfjg

òreuhj hbkf

,

jgl tcf qufstbog bs wcftcfr bt bs possbmhf to jssbig Moohfjg vjhufs to tcf vjrbjmhfs x, y, z,

so tcjt tcf òreuhj fvjhujtfs to \]VF (Frbaksog, =>:1). Mfhow bs j hbst o` soef o` tcf eost

popuhjr gp-aoephftf promhfes aogtjbgfl bg jg bejif bhhustrjtbgi tcf cbfrjracy o`

rfluatbogs to ibvf j pbaturf o` wcfrf tcf \Q^ stjgls4

\cbs stuly wbhh got io bgto tcf lftjbhs o` provbgi gp-aoephftfgfss o` promhfes. ]fvbfwfl

hbtfrjturf, suac js Lrfwfs (=>:1), Mrblif (=>>6m), jgl (Iopjhjkrbscgjg, =>>0) ajg mfaogsuhtfl òr eftcols òr proo` o` gp-aoephftfgfss.

=.=.>6)


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::

bs tcf sjef brrfspfatbvf o` lbrfatbog, wcbac bg tcf jsyeeftrba, tcjt bs got j rfqubrfefgt. \o

bhhustrjtf irjpcbajhhy, mfhow bs j 0-abty \Q^ jgl j possbmhf routf tcjt ajg mf tjkfg4

\cf rjtf o` irowtc o` tcf lb``bauhty o` sohvbgi tcf \Q^ wbtc bgput sbzf bs rfjhhy pcfgoefgjh.Mfhow bs j tjmhf wcbac scows tcf proirfssbog bg guemfr o` possbmhf tours, guemfr o` flifs

jgl fstbejtfl tbef to sohvf usbgi mrutf-`oraf, wbtc tcf BME ]ojlruggfr supfraoeputfr js

fxphjbgfl bg sfatbog :.:.

J

M

A

@

0

=

1 =

3

sfaogls

=> :2> 0>,3==,66>,=>> > 1:>,==>,300,0:2,9:2,03>,>>> 0 yfjrs

1> ,299,=9:,3>3,>>>,>>> 26,26=,>2: yfjrs

16 626 :,=>9,>,21>,,>>>,>>> 1,=>1,2>2, :>,:23,2,623,9=:,092,1=>,:,302,26:,>>,>>> ==:,10>,92=,93=,2:0,026,220,::1 yfjrs

:,1=2,:16,939,129,1>9,2>,>>>,>>>=3,3=:,31>,996,

yfjrs

6> :==61>:9,:11,93>,3:,00>,0>,>>>,>>>0,0>:,:01,6,901,=1:,2>:,9>>,>>>,>>>=,6>6,:0,102,1>>,>>>,>>>,>>>:,6>6,>>2,


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\cf stuly o` tcf \Q^ cjs iogf mfyogl dust tcf ahjssbajh promhfe, bt mfbgi usfl js j

òugljtbog òr stulybgi ifgfrjh eftcols tcjt jrf jpphbfl to j wblf rjgif o` optbebzjtbog

promhfes. Js cjs mffg stjtfl bg acjptfr ogf, ejtcfejtbajh promhfes rfhjtfl to tcf \Q^ wfrf

stulbfl rbict `roe tcf :9tcAfgtury. Qbr Ybhhbje ]owjg Cjebhtog arfjtfl tcf Baosbjg Ijef

bg :369, wcbac bgvohvfl j pfimojrl wbtc twfgty cohfs bg bt, rfqubrbgi fjac cohf to mf vbsbtfl

oghy ogaf, go pjtc mftwffg two cohfs mfbgi vbsbtfl eorf tcjg ogaf, jgl tcf fglbgi pobgt

mfbgi tcf sjef js tcf stjrtbgi pobgt. Cowfvfr, \Q^ bg bts ifgfrjh òre wjs `brst stulbfl bg tcf

hjtf :2=>s to fjrhy :21>s my Kjrh Efgifr (Qjbyfl, =>:=). Bg :29=, ]bacjrl E. Kjrp

scowfl tcjt tcf Cjebhtogbjg ayahf, wcbac rfluafs to tcf \Q^, bs gp-aoephftf (Ejrlbj, =>:>).

\cbs bs fvfg omvbous, aogsblfrbgi tcf `jat tcjt sohvbgi tcf \Q^ my fguefrjtbgi jhh possbmhf

routfs wouhl bgvohvf fguefrjtbgi ((g-:)/=)! routfs. Ybtc tcf bgtfgsf rfsfjrac lfvotfl to tcf

\Q^, cowfvfr, bgarfjsbgihy hjrifr bgstjgafs o` tcf promhfe cjvf mffg sohvfl. J mrfjklowg

o` tcf ebhfstogfs bg pro``frbgi sohutbog to bgstjgafs o` tcf \Q^ jaaorlbgi to (Aook, =>>9) bs

prfsfgtfl mfhow4

Ljgtzbi, @uhkfrsog, jgl Docgsog pumhbscfl j lfsarbptbog o` j eftcol òr sohvbgi tcf \Q^ jgl

bhhustrjtfl tcf powfr o` tcbs eftcol my sohvbgi jg bgstjgaf wbtc


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:::), jgl cjs provfg fxtrfefhy lb``bauht to jgswfr òrsabfgtbsts tcf worhl ovfr, aogsblfrbgi tcf promhfe btsfh` bs bg G^ (Ljsiuptj, ^jpjlbebtrbou, &

Pjzbrjgb, =>>0)! Mfajusf o` tcf rfluatbog propfrty o` gp-aoephftf promhfes, provbgi tcjt ^

5 G^ (b.f. `bglbgi j pohygoebjh sohutbog to jg gp-aoephftf promhfe) wouhl bephy tcjt jhh gp-

aoephftf promhfes cjvf mfaoef sohvjmhf bg pohygoebjh tbef. \cf bephbajtbogs o` tcbs bg tcf

rfjh worhl rfjac `jr jgl wblf, `roe pumhba-kfy fgaryptbog `jbhbgi, to tcf possbmbhbty o`

jrtb`babjh systfes beprovbgi jgl mfaoebgi aoephftfhy out o` cuejg aogtroh, (possbmhy)

ovfrtcrowbgi ejgkbgl jgl tjkbgi aogtroh o` tcf worhl (Aook, =>::)! Cowfvfr, tcf popuhjropbgbog bg tcf sabfgtb`ba aoeeugbty bs tcjt ^ G^, jgl tcbs jssueptbog fvfg òres j mjsbs òr

ejgy tcforfes jgl sohutbogs toljy.

=.=.0 Vgsohvjmhf ^romhfes

Bgtfrfstbgihy, tcfrf jrf soef promhfes wcbac cjvf mffg provfg to mf aoephftfhy ugsohvjmhf,

suac js tcf ejtcfejtbajh vfrsbog o` QJ\, wcbac bs, ibvfg j pohygoebjh fqujtbog bg ejgy

vjrbjmhfs, f.i.

= 9 5 0,

jrf tcfrf bgtfifr vjhufs o`x, y, z tcjt sjtbs`y bt? \cf `brst o` tcbs kbgl wjs lbsaovfrfl bg :210

my Jhjg \urbgi (Ljsiuptj, ^jpjlbebtrbou, & Pjzbrjgb, =>>0).

=.=.9

Qohvbgi Cjrl ^romhfes

Js cjs mffg fjrhbfr stjtfl, jgl my our lf`bgbtbog o` tcf tfre sohvf or sohutbog, tcf ahjss o`

promhfes uglfr stuly ajggot mf sohvfl fxjathy òr gow. Cowfvfr, tcfsf jrf promhfes wcbacwf `jaf bg our fvfr-ljy hbvfs, jgl wcbac ajggot mf jvoblfl. Cow tcfg ajg wf ovfraoef tcbs

mjrrbfr? \cf jvjbhjmhf possbmbhbtbfs bgahulf4

Qohvf sbephfr or eorf spfab`ba vfrsbogs or bgstjgafs o` tcf promhfe. Yf aouhl sffk to

rf-eolfh tcf promhfe, or vjry bt j hbtthf my ejkbgi soef jssueptbogs jgl aogstrjbgbgi

soef vjrbjmhfs, tcfrfmy trjgsòrebgi bt bgto j òre wcbac bs sohvjmhf bg rfjsogjmhf

tbef.


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Vsf j pohygoebjh-tbef promjmbhbstba jhiorbtce, b.f. ogf tcjt cjs j cbic promjmbhbty o`

ibvbgi tcf rbict jgswfr, jgl bg rfjsogjmhf tbef.

@or optbebzjtbog promhfes, usf j pohygoebjh-tbef jpproxbejtbog jhiorbtce, ogf

wcbac bs got iujrjgtffl to `bgl tcf mfst jgswfr, mut wbhh ibvf ogf wcbac bssjtbs`jatorbhy ahosf. \cfsf kbgls o` jhiorbtces jrf ajhhfl cfurbstbas, jgl ^QO `jhhs bgto

tcbs ajtfiory (Mrblif, =>>6j).

=.1 Cfurbstbas/Eftj-Cfurbstbas

\cf tfre cfurbstba cjs mffg wblfhy usfl my ejgy tcforftbajh aoeputfr sabfgtbsts ovfr tcf

yfjrs, wbtc ejgy o` tcfe ibvbgi vjrybgi lf`bgbtbogs o` tcf tfre. Jaaorlbgi to ]oejgyabj &

^fhhftbfr (:236), tcf aogafpt o` cfurbstbas stjrtfl jppfjrbgi bg JB hbtfrjturf bg tcf fjrhy

:26>s, jgl mfajef popuhjr my tcf fjrhy :20>s. \cf worl cfurbstba aoefs `roe tcf jgabfgt

Irffk worl cfurbskfbg, jgl tcf Hjtbg worl cfurbstbaus wcbac efjg to `bgl out, lbsaovfr.

Cowfvfr, bg tcf aogtfxt tcforftbajh aoeputfr sabfgaf, tcbs lf`bgbtbog wouhl sffe too

sbephbstba. \cf work o` ]oejgyabj & ^fhhftbfr (:236) bg stulybgi tcf lf`bgbtbogs o` tcbs

worl bs j ahjssba jgl aoeprfcfgsbvf ogf, jgl ogf og wcbac wf wouhl hbkf to rfhy to ibvf j

òrejh lf`bgbtbog o` tcf aogafpt. Jaaorlbgi to tcfe, tcf aogafpt cjs, mjsfl og tcf vjrbous

lf`bgbtbogs pro``frfl, mffg jssoabjtfl wbtc tcf òhhowbgi lbefgsbogs o` efjgbgi4

ugafrtjbgty o` outaoef, mjsbs bg bgaoephftf kgowhflif, beprovfefgt o` pfròrejgaf jgliubljgaf o` lfabsbog ejkbgi. J`tfr rfvbfwbgi vjrbous lf`bgbtbogs, tcfy ijvf j òrejh

lf`bgbtbog wcbac cjreogbzfs tcfsf òur lbefgsbogs o` efjgbgi4

J cfurbstba bg jrtb`babjh bgtfhhbifgaf bs jgy lfvbaf, mf bt j proirje, ruhf, pbfaf o`kgowhflif, fta., wcbac ogf bs got fgtbrfhy aog`blfgt wbhh mf usfùh bg provblbgi j

prjatbajh sohutbog, mut wcbac ogf cjs rfjsog to mfhbfvf wbhh mf usfùh, jgl wcbac[ejy mfU jllfl to j promhfe sohvbgi systfe bg fxpfatjtbog tcjt og jvfrjif tcf

pfròrejgaf wbhh beprovf.

Bg sbephf tfres, j cfurbstba jttfepts to sohvf j prjatbajh promhfe tcrouic rjgloe tcouic

bgtfhhbifgt ruhf-mjsfl lfabsbogs wcbac jrf got iujrjgtffl to provblf tcf mfst sohutbog, mut

cjvf j vfry iool acjgaf o` provblbgi j sjtbs`jatory ogf. Js òr tcf tfre eftj-cfurbstba,

wcbhf soef sabfgtbsts usf bt bgtfracjgifjmhy wbtc tcf tfre cfurbstba (Rjgi S.-Q. , Gjturf-

Bgspbrfl Eftjcfurbstba Jhiorbtces, =>:>m), otcfrs prf`fr to lf`bgf bt js j sacfef wcbac

iublfs sumorlbgjtf cfurbstbas bg ejkbgi eorf bgtfhhbifgt lfabsbogs wcbac hfjl to mfttfr

fxphorjtbog jgl fxphobtjtbog, fvfgtujhhy hfjlbgi to (gfjr-)optbejh sohutbogs, jgl wcbac jrf


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got promhfe-spfab`ba (Mhue & ]ohb, =>>1). Bg j hjy ejgs tfres, j eftjcfurbstba bs mjsbajhhy j

eorf bgtfhhbifgt cfurbstba.

=.1.: Ahjssb`bajtbogs o` Eftj-Cfurbstbas

Lb``frfgt wjys fxbst tcrouic wcbac eftj-cfurbstbas ajg mf ahjssb`bfl. Jaaorlbgi to Mhue &

]ohb (=>>1), tcf òhhowbgi ejbg ahjssb`bajtbogs o` eftj-cfurbstbas fxbst4

Gjturf-bgspbrfl vs. gog-gjturf bgspbrfl4 \cbs bs j eftcol o` ahjssb`bajtbog mjsfl og tcf orbibg

o` tcf eftj-cfurbstba, b.f. tcosf wcbac lrjw tcfbr bgspbrjtbog `roe gjturjh pcfgoefgj, hbkf

Fvohutbogjry Jhiorbtces (Ajhijrb, Aorjy, Cfrtz, Komhfr, & Kougfg, :222; Roussf`, Qjbt, &

Jlbacf, =>>:) jgl Qwjre Bgtfhhbifgaf (Kfggfly & Fmfrcjrt, =>>:), jgl tcosf wco jrf got

bgspbrfl my gjturf, hbkf \jmu Qfjrac (Ifglrfju & ôtvbg, =>:>) jgl Btfrjtfl Hoajh Qfjrac

(Hourfgo, Ejrtbg, & Qttzhf, =>:>).

ôpuhjtbog-mjsfl vs. sbgihf pobgt sfjrac4 ôpuhjtbog-mjsfl eftj-cfurbstbas usf eorf tcjg ogf

ajglbljtf sohutbog to sbeuhtjgfoushy fxphorf tcf sfjrac spjaf (Qwjre Bgtfhhbifgaf (QB),

Ifgftba Jhiorbtces (IJ)), wcbhf sbgihf pobgt sfjrac ogfs usf oghy ogf sohutbog jt j tbef, jgl

jrf ajhhfl trjdfatory eftcols. Fxjephfs jrf \jmu Qfjrac (\Q), Btfrjtfl Hoajh Qfjrac (BHQ)

jgl Pjrbjmhf Gfbicmorcool Qfjrac (PGQ).

Ogf vs. Pjrbous Gfbicmorcool Qtruaturfs4 @or eftj-cfurbstbas hbkf QB, BHQ jgl \Q, tcf`btgfss hjglsajpf topohoiy rfejbgs tcf sjef tcrouicout tcf aoursf o` tcf fxfautbog o` tcfbr

proafssfs. Otcfrs hbkf PGQ usf j sft o` gfbicmorcool struaturfs wcbac jhhow lb``frfgt `btgfss

hjglsajpfs to mf fxphorfl.

Efeory usjif vs. efeory-hfss eftcols4 Qoef eftj-cfurbstbas ejkf tcf gfxt lfabsbog mjsfl

og jg jaaueuhjtbog o` ljtj lurbgi tcf prfaflbgi lfabsbogs, wcbhf otcfrs oghy tjkf lfabsbogs

mjsfl og tcf aurrfgt stjtf o` tcf sfjrac proafss.

@roe tcf jmovf ahjssb`bajtbogs, bt wbhh mf sffg tcjt Qwjre Bgtfhhbifgaf, wcbac bs tcf òaus o`

tcbs rfsfjrac, bs j popuhjtbog-mjsfl, gjturf-bgspbrfl efeory usbgi eftjcfurbstba wbtc j ogf-

gfbicmorcool eftj-cfurbstbas.

=.


27/128

:9

Qwjre Bgtfhhbifgaf bs jg bggovjtbvf lbstrbmutfl bgtfhhbifgt pjrjlbie òr sohvbgioptbebzjtbog promhfes tcjt orbibgjhhy took bts bgspbrjtbog `roe mbohoibajh fxjephfs

my swjrebgi, `hoakbgi jgl cfrlbgi pcfgoefgj bg vfrtfmrjtfs. (Jmrjcje, Iuo, &Hbu, =>>0).

Ejgy jgbejhs bg gjturf hbvf bg aohogbfs, swjres or cfrls, mfajusf tcfy tfgl cjvf j cbicfracjgaf o` survbvjh wcfg tcfy hbvf toiftcfr tcjg wcfg tcfy hbvf jhogf. @or fxjephf, j hogf

`bsc bs jg fjsbfr tjrift to j prfljtor tcjg j sacooh o` `bsc, jgl tcf sacooh o` `bsc cjvf j mfttfr

pfrafptbog o` tcf fgvbrogefgt, jgl so ajg jvobl prfljtors mfttfr, jgl `bglbgi o` òol bs euac

fjsbfr. Jhh tcbs cjppfgs wbtcout jgy lfsbigjtfl cfjl `bsc lbrfatbgi jhh tcf otcfr `bsc bg tcf

sacooh. \cjt bs fefrifgaf. Aoeeugbajtbog mftwffg bglbvblujhs bs o` uteost beportjgaf bg

tcfsf swjre systfes. \cbs aoeeugbajtbog ajg mf fbtcfr lbrfat js bg julbtory or vbsujh, suac

js bg mbrl `hoaks jgl tcf wjiihf ljgaf o` cogfy mffs wcbac ajg mf lbrfathy omsfrvfl jgl

uglfrstool my otcfr mffs, or bglbrfat, wcfrf aoeeugbajtbog bs ajrrbfl out my jhtfrbgi tcf

fgvbrogefgt (stbiefriy). Jg fxjephf o` stbiefriy bs tcjt sffg wcfrf jgts aoeeugbajtf

tcrouic tcf lfposbtbog o` j acfebajh ajhhfl pcfroeogf wcbac tcfy ajg sefhh (Qulcoht, =>>3;

Jcefl & Ihjsiow, =>:=).

\cf kfy (jgl eost bgtfrfstbgi, bg our opbgbog) acjrjatfrbstba o` swjre bgtfhhbifgaf bs

fefrifgaf, tcjt bs, tcjt vfry sbephf jifgts, ajrrybgi out vfry sbephf bgtfrjatbogs ajg proluaf

aoephfx orijgbzjtbog jgl sohvf aoephfx promhfes. \cbs bs wcjt fefrifgaf fgtjbhs, jgl tcbsacjrjatfrbstba bs lfsarbmfl bg lftjbh bg tcf òhhowbgi lfsarbptbog (Hjkkjkuhj, Qwjpgj, ]jeyj,

& Ejcftcj, =>:


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:3

òreuhjtfl `bvf mjsba prbgabphfs o` swjre bgtfhhbifgaf wcbac cfhp bg aogarftfhy lf`bgbgi wcjt

swjre bgtfhhbifgaf rfjhhy bs4

\cf prbgabphf o` proxbebty4 \cf swjre popuhjtbog scouhl mf jmhf to ajrry out sbephf

spjaf jgl tbef aoeputjtbogs. \cjt bs, bglbvblujh jifgts scouhl ajpjmhf o` soef òre

o` bgtfrjatbog jgl aoeeugbajtbog mftwffg tcfesfhvfs.

\cf qujhbtyprbgabphf4 \cf popuhjtbog scouhl mf jmhf to rfspogl to spfab`bfl/spfab`ba

qujhbty `jators bg tcf fgvbrogefgt, suac js òol jgl sj`fty, wcbac jppjrfgthy scouhl

iublf tcf popuhjtbogsmfcjvbour.

\cf prbgabphf o` lbvfrsf rfspogsf4 \cf popuhjtbog scouhl got aoeebt bts jatbvbty

jhogi fxafssbvfhy gjrrow acjggfhs.

\cf stjmbhbty prbgabphf4 \cf popuhjtbog scouhl got acjgif bts eolf o` mfcjvbourfvfry tbef tcf fgvbrogefgt acjgifs.

\cf prbgabphf o` jljptjmbhbty4 \cf popuhjtbog scouhl mf jmhf to acjgif mfcjvbour

eolf wcfg bts wortc tcf aoeputjtbogjh prbaf.

Qwjre bgtfhhbifgaf eftj-cfurbstbas cjvf mffg òugl to cjvf ejgy jlvjgtjifs ovfr

trjlbtbogjh, lftfrebgbstba eftcols jgl jhiorbtces, jgl js j rfsuht, cjvf mffg jpphbfl bg jheost

fvfry jrfj o` sabfgaf, fgibgffrbgi jgl bglustry (Rjgi S.-Q. , Qwjre-Mjsfl Eftjcfurbstba

Jhiorbtces jgl Go-@rff-Hugac \cforfes, =>:=). Pjrbous swjre bgtfhhbifgaf-mjsfl eftj-

cfurbstbas cjvf mffg lfvfhopfl ovfr tcf yfjrs sbgaf tcf bgtroluatbog o` tcf tfre my I. Mfgb

jgl D. Yjgi bg :232 (Jcefl & Ihjsiow, =>:=). Qoef jrf Jgt Aohogy Optbebzjtbog (JAO)

(Lorbio, :22=); ^jrtbahf Qwjre Optbebzjtbog (^QO) (Kfggfly & Fmfrcfrt, :226); Jrtb`babjh

Mff Aohogy (JMA) (Kjrjmoij, =>>6); @brf`hy Jhiorbtce (@J) (Rjgi S.-Q. , =>>2); Mff

Aohogy Optbebzjtbog (\folorovba, Huaba, Ejrkovba, & Lfhh' Orao, =>>0); Auakoo Qfjrac

(AQ) (Rjgi & Lfm, =>>2); jgl Mjt-Bgspbrfl Jhiorbtce (MJ) (Rjgi S.-Q. , =>:>j). Yf wbhh

ibvf j mrbf` lfsarbptbog o` JAO jgl MAO mfòrf ^QO, tcf ejbg òaus o` tcbs rfsfjrac, wbhh mf

lbsaussfl.

=.:>m). \cf blfj mfcbgl bt aoefs `roe tcf wjy jgts `bgl tcf scortfst wjy to j

òol lfposbt. Jt tcf bgbtbjh stjif, tcfy jhh wjglfr jmout rjgloehy hookbgi òr òol. Js tcfywjglfr jrougl, tcfy lrop pcfroeogf og tcf irougl. Ogaf jgy o` tcfe `bgls òol, bt rfturgs


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:2

to tcf gfst. Bt tcfg rfturgs jijbg to tcjt sjef òol souraf, mjsfl og tcf prbgabphf tcjt jgts

jhwjys prf`fr to òhhow tcf safgt o` pcfroeogf, jgl tcf strogifr tcf safgt, tcf cbicfr tcf

acjgaf tcjt tcfy wbhh òhhow bt. Cowfvfr, my tcf gjturf o` pcfroeogf, bt fvjporjtfs ovfr tbef.

\cfrfòrf, ovfr tbef, b` lb``frfgt jhtfrgjtf routfs cjvf mffg òugl to tcf òol souraf, tcf ogaf

wcbac bs ahosfst wbhh ift sfhfatfl my fvfry jgt, mfajusf bts pcfroeogf aogafgtrjtbog bgarfjsfs

`jstfr tcjg jhh tcf otcfrs (Lorbio, :22=). Bg bephfefgtbgi tcbs systfe jrtb`babjhhy, jrtb`babjh

jgts aogstruat sohutbogs òr j ibvfg AO promhfe my j spfab`bfl aogstruatbvf eftcol.

\ypbajhhy, j sohutbog bs aogstruatfl my ejkbgi j sfqufgaf o` promjmbhbstba lfabsbog wcfrf

fvfry lfabsbog fxtfgls j pjrtbjh sohutbog my jllbgi j gfw sohutbog aoepogfgt ugtbh j

aoephftf sohutbog bs lfrbvfl. Ogaf j aoephftf sohutbog bs lfrbvfl, bt bs fvjhujtfl to sff b` bt bs

j iool ogf. B` bt bs, tcf flifs (b.f. tcf sumaoepogfgts tcjt ejlf up tcf sohutbog) o` tcf pjtc

tcjt hfl to bt jrf ejrkfl wbtc jrtb`babjh pcfroeogf. \cf pcfroeogf iublfs jgts bg tcf gfxt

btfrjtbog towjrls pjtcs tcjt ajg hfjl to iool sohutbogs. Lurbgi jg upljtf o` tcf pcfroeogf

vjhufs soef o` tcf pcfroeogf fvjporjtfs, to prfvfgt ohlfr btfrjtbogs `roe j``fatbgi

sumsfqufgt ogfs òr too hogi, jgl bg hbgf wbtc tcf eolfh o` gjturjh jgts. \cf proafss

aogtbgufs ugtbh j spfab`bfl tfrebgjtbog arbtfrbj bs eft (Efrkhf & Ebllfglor`, =>>6).

=.:=) jgl \folorovba, Huaba, Ejrkovba, & Lfhh' Orao (=>>0) òr lftjbhs.

=.:>). \cf


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=>

^QO eftjcfurbstba jbes jt sohvbgi promhfes o` j typf rf`frrfl to js Aoembgjtorbjh

Optbebzjtbog (AO) promhfes. Bg sbephf tfres, j Aoembgjtorbjh Optbebzjtbog promhfe bs ogf

wcbac bs ejlf o` jg omdfatbvf ùgatbog wcbac bs to mf ebgbebzfl, j sft o` possbmhf/`fjsbmhf

sohutbogs wcbac bs ajhhfl tcf sfjrac spjaf, wcfrf fjac sohutbog cjs jg jssoabjtfl `btgfss vjhuf

omtjbgfl wcfg bt bs pjssfl bgto tcf omdfatbvf ùgatbog, jgl tcf sohutbogs cjvf soef

aogstrjbgts mbglbgi tcfe. \cfrfòrf, sohvbgi j AO promhfe bgvohvfs `bglbgi tcjt sohutbog

jeogi tcf sft o` `fjsbmhf sohutbogs wcbac cjs tcf ebgbeue omdfatbvf ùgatbog vjhuf. \cbs

sohutbog bs ajhhfl tcf ihomjh optbejh sohutbog. Fxjephfs o` AO promhfes jrf tcf \Q^, tcf

Zujlrjtba Jssbigefgt ^romhfe, \beftjmhbgi jgl Qacfluhbgi ^romhfes. @or eorf lftjbhs, jgl

j eorf ejtcfejtbajhhy rbiorous lf`bgbtbog, sff Mhue & ]ohb (=>>1).

Qwjre

\cf tfre swjre sbephy rf`frs to j sft o` pjrtbahfs. Jaaorlbgi to Kfggfly jgl Fmfrcfrt, tcf

arfjtors o` ^QO, tcfbr eotbvjtbog mfcbgl ajhhbgi tcf aohhfatbog o` pjrtbahfs j swjre bs

sbephy tcf `jat tcjt tcf two-lbefgsbogjh phots wf usfl to wjtac tcf jhiorbtces pfròre

afjsfl to hook euac hbkf mbrl `hoaks or `bsc sacoohs jgl stjrtfl hookbgi eorf hbkf swjres o`

eosqubtos. \cf gjef ajef js sbephy js tcjt. (Kfggfly & Fmfrcjrt, =>>:). Jgl wf hfjvf

tcf fxphjgjtbog jt tcjt!

^jrtbahf

Jaaorlbgi to Kfggfly & Fmfrcjrt (=>>:), tcf tfre pjrtbahf ajef mfajusf popuhjtbog

efemfrs o` tcf ^QO jrf ejsshfss jgl vohuefhfss ejtcfejtbajh jmstrjatbogs jgl wouhl mf

ajhhfl pobgts b` tcfy stjyfl stbhh; vfhoabtbfs jgl jaafhfrjtbogs jrf eorf jpproprbjtfhy jpphbfl

to pjrtbahfs, fvfg b` fjac bs lf`bgfl to cjvf jrmbtrjrbhy sejhh ejss jgl vohuef.Qo, wcjt bs j

pjrtbahf? Jaaorlbgi to (Ahfra E. , =>:=), j pjrtbahf bs ejlf o` tcf ohhowbgi4

- j posbtbog, x, bgsblf tcf sfjrac spjaf

- tcf `btgfss vjhuf jt tcbs posbtbog

- j vfhoabty, v, wcbac wbhh mf usfl to aoeputf tcf gfxt posbtbog

- j efeory o` tcf mfst posbtbog bt cjs òugl so `jr, ajhhfl tcf pfrsogjh mfst or pmfst

- tcf `btgfss vjhuf o` bts pfrsogjh mfst

Gow, òr j aoephftf lf`bgbtbog o` ^QO4 bt bs j eftjcfurbstba tcjt fephoys j popuhjtbog o`

pjrtbahfs ajhhfl j swjre wcbac `hbfs rjgloehy tcrouic tcf sfjrac spjaf, fjac pjrtbahfejbgtjbgbgi j efeory o` bts mfst posbtbog so `jr, tcf mfst posbtbog jttjbgfl my jgy o` tcf


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=:

pjrtbahfs mfbgi jhso kfpt bg efeory, suac tcjt tcf lbrfatbog jgl spffl o` `hbict o` fjac

pjrtbahf bs j stoacjstba jvfrjif o` bts pfrsogjh mfst jgl tcf ihomjh mfst, wcbhf tcf fgtbrf

swjres eovfefgt bs mbjsfl towjrls tcf eost proebsbgi rfibogs lbsaovfrfl so `jr.

\cf upljtf fqujtbogs usfl òr tcf ahjssbajh (orbibgjh) ^QO jrf js òhhows4

+ 5

(

) (:)

+ 5

+.(=)

Ycfrf

5 :, =, ,

5 rjtf o` posbtbog acjgif o` tcf btc pjrtbahf bg tcf ltc lbefgsbog. \cf bgbtbjh vfhoabty, ,

sfrvfs js j efjgs o` bgfrtbjh eovfefgt to tcf pjrtbahf my tjkbgi bts prfvbous posbtbog scb`t

bgto aogsblfrjtbog. Bg `jat, b` bt cjppfgs tcjt j pjrtbahf `bgls j posbtbog wcosf `btgfss vjhuf bs

howfr tcjt o` jgy o` tcf posbtbogs lbsaovfrfl so `jr my tcf wcohf swjre, tcjt posbtbog

mfaoefs bts pfrsogjh mfst, js wfhh js tcf ihomjh mfst. Bg tcjt ajsf, tcf two rjgloe tfres bg

fq(:) wouhl mfaoef zfro, jgl tcf pjrtbahf wouhl cjvf to rfejbg jt tcf sjef posbtbog òr

soef tbef, ugtbh jgotcfr pjrtbahf lbsaovfrs j mfttfr posbtbog, cfgaf mfbgi cbglfrfl `roe

aogtbgubgi bts sfjrac. \cbs wouhl rfluaf tcf sfjrac ajpjmbhbty o` tcf swjre my ogf, js jt jgypobgt bg tbef lurbgi tcf sfjrac proafss, ogf o` tcf pjrtbahfs bs jt rfst. Cowfvfr, wbtc tcf

bgbtbjh vfhoabty tfre, tcbs sbtujtbog bs jvoblfl.

5 posbtbog o` tcf btc pjrtbahf bg tcf ltc lbefgsbog, jgl gorejhhy rfprfsfgts tcf pjrtbahf

btsfh`, js j vfator o` bts posbtbogs bg jhh lbefgsbogs o` tcf promhfe spjaf

a:5 aoigbtbvf pjrjeftfr wcbac aogtrohs cow euac tcf pjrtbahfs fxpfrbfgaf aogtrbmutfs to

tcf gfw posbtbog. \cf vjhufs o` a: jgl a= wfrf orbibgjhhy motc sft to =, mut òr irfjtfr aogtrohog sfjrac jmbhbty, rfafgt vfrsbogs o` ^QO usf lb``frfgt vjhufs, wbtc a: + a= 5


32/128

==

a= 5 soabjh pjrjeftfr, aogtrohs cow euac tcf pjrtbahf eovfs towjrls tcf ihomjh mfst, js

fxphjbgfl jmovf.

r: & r= 5 rjgloe vjrbjmhfs ugbòrehy lbstrbmutfl wbtcbg [>,:U, tcf gfafssjry fhfefgt o`

stoacjstbabty.

5 tcf mfst posbtbog lbsaovfrfl so `jr my tcf btc pjrtbahf

5 tcf mfst posbtbog lbsaovfrfl my jgy o` tcf pjrtbahfs bg tcf swjre

5btfrjtbog aougtfr (Jcefl & Ihjsiow, =>:=; ^jrsopouhos & Prjcjtbs, =>:>).

Bg sueejry, tcf jhiorbtce ajg mf prfsfgtfl js òhhows4

Jhiorbtce B4 ^jrtbahf Qwjre Optbebzjtbog

:4 ]jgloehy bgbtbjhbzf swjre

=4 wcbhf(tfrebgjtbog arbtfrbj got eft) {

14 òr(t 5 :; t85swjresbzf; t++) {

4 b` (`(pmfstt)8 `(imfst)) imfst 5 pmfstt

::4 }

:=4 }

Ybtc tbef, js ^QO wjs jpphbfl to eorf promhfes, lf`babfgabfs mfijg to fefrif. Pfhoabty

ahjepbgi cjl to mf bgtroluafl mfajusf o` tcf tfglfgay o` tcf swjre to lbvfrif fglhfsshy jgl

wbtcbg j scort wcbhf, sbgaf tcf bgbtbjh vfhoabty tfre fgsurfl tcjt tcf vfhoabty wjs jhwjys

bgarfjsbgi. J spfab`ba vfhoabty tcrfscohl, vejx bs lf`bgfl my tcf usfr, jgl lfabsbogs tjkfg

wcbac wbhh kffp tcf vfhoabty js ahosf js bt ajg mf to tcf vfhoabty mougl wcfg tcf tcrfscohl bs

fxafflfl. Qumsfqufgthy, jhtcouic tcf promhfe o` vfhoabty fxphosbog wjs sohvfl, tcf bgjmbhbty

o` tcf jhiorbtce to fxphobt proebsbgi rfibogs, fspfabjhhy towjrls tcf fgl o` tcf fxfautbog

mfajef fvblfgt. \cjt bs, tcf aogvfrifgaf powfr o` tcf jhiorbtce wjs bgjlfqujtf. \cbs jhso

wjs j rfsuht o` tcf bgbtbjh vfhoabty tfre. Jirffl, jt tcf fjrhy stjifs o` tcf jhiorbtce, bt wjs

lfsbrjmhf tcjt tcf vfhoabtbfs mf cbic to fgaourjif ejssbvf fxphorjtbog. Cowfvfr, towjrls tcf


33/128

=1

fgl o` tcf proafss, tcfrf wjs j gffl to aogtroh tcf vfhoabtbfs so tcjt tcfy mfaoef sejhh to

fgaourjif fxphobtjtbog jrougl rfibogs jhrfjly òugl to mf proebsbgi. \o jllrfss tcbs

acjhhfgif, Qcb & Fmfrcfrt (:223) bgtroluafl j gfw ^QO vjrbjgt. Bg tcf gfw vjrbjgt, j gfw

pjrjeftfr ajhhfl bgfrtbj wfbict wjs bgtroluafl bgto fqujtbog (:), jgl wjs sumsfqufgthy

ifgfrjhhy bephfefgtfl suac tcjt bt wouhl stjrt jt j vjhuf shbicthy irfjtfr tcjg : jgl wbtc

tbef, hbgfjrhy lfarfjsf towjrls zfro, or prf`frjmhy, j strbathy posbtbvf howfr mougl suac js >.:

(to prfvfgt tcf prfvbous vfhoabty `roe vjgbscbgi). \cf gfw ^QO vjrbjgt bs ibvfg mfhow4

+ 5

(

) (1)

+ 5

+.(>=) (^jrsopouhos & Prjcjtbs,

=>:>).

=.) lfvfhopfl j govfh sft-mjsfl ^QO eftcol òr Lbsarftf Optbebzjtbog

promhfes, wcbac tcfy ajhhfl Qft-mjsfl ^QO (Q-^QO) jbefl jt usbgi j sft-mjsfl

rfprfsfgtjtbog sacfef to acjrjatfrbzf tcf sfjrac spjaf o` aogtbgujh optbebzjtbog promhfes,

jgl wcbac aouhl mf usfl to fxtfgl tcf vjrbous ^QO vjrbjgts to tcfbr lbsarftf vfrsbogs. \cfy

usfl tcfbr jhiorbtce to lfvfhop j lbsarftf vfrsbog o` tcf Aoeprfcfgsbvf Hfjrgbgi

^QO(AH^QO) proposfl my Hbjgi ft jh (=>>0). \cfy fvjhujtfl tcf jhiorbtce my usbgi bt to

sohvf soef mfgacejrk \Q^ promhfes jgl aoepjrbgi tcf rfsuhts wbtc tcosf o` motc fxbstbgi

^QO-mjsfl jpprojacfs jgl soef eftj-cfurbstba jhiorbtces. Jt tcf fgl, tcfy òugl tcfbr

jhiorbtce to mf proebsbgi. Jhso, Ej ft jh (=>:1) proposfl jg f``babfgt lbsarftf ^QO

jhiorbtce bg orlfr to jacbfvf optbebzjtbog bg lbsarftf spjaf. \cbs wjs j eolb`bajtbog o` tcf

work o` Zu, Quijgtcjg, & Ljs (=>:1) ajhhfl Hoajhhy Bgòrefl ^jrtbahf Qwjre (HB^Q) mjsfl

og sft-tcfory. \cfy ajhhfl tcf jhiorbtce Qft-mjsfl HB^Q (Q-HB^Q). \cfy tcfg aoepjrfl tcf

pfròrejgaf o` Q-HB^Q og soef mfgacejrk \Q^ bgstjgafs wbtc tcjt o` Q-AH^QO, jgl òugl

tcf rfsuhts to mf proebsbgi, fspfabjhhy bg tcf ajsf o` hjrif sajhf promhfes. J`jq & Qjbgb


34/128

=:=) jhso prfsfgtfl j Lbsarftf ^jrtbahf Qwjre Optbebzjtbog (L^QO) tfacgbquf to sohvf

irjpc-mjsfl \Q^, tfstfl bts aogvfrifgaf, jgl òugl tcf rfsuht `jvourjmhf. Yu ft jh (=>:1)

proposfl j lbsarftf ^QO (L^QO) jhiorbtce wbtc saout pjrtbahfs to sohvf tcf promhfe o`

sfhfatbgi ejtfrbjhs òr j hbmrjry bg orlfr to ejxbebzf tcf jvfrjif prf`frfgaf jgl tcf mulift

fxfautbog rjtf uglfr soef prjatbajh rfstrbatbogs bgahulbgi lfpjrtefgtjh mulift, hbebtjtbog o`

guemfr o` ejtfrbjhs bg fjac ajtfiory jgl fjac hjgiujif. \cfy pfròrefl j sfrbfs o`

aoeputjtbogjh fxpfrbefgts, stjtbstbajhhy jgjhyzfl tcf rfsuhts jgl aogahulfl tcjt tcfbr

proposfl L^QO jhiorbtce bs jg f``fatbvf jpprojac to tcf promhfe. Otcfr sacfefs wcbac

cjvf mffg lfvfhopfl to sohvf tcf \Q^ usbgi ^QO bgahulf @jgi, Acfg, & Hbu (=>>9), @jg

(=>:>), Iohlmjri, Iohlmjri, & Ibvjgjhlo (=>>3), \jsiftbrfg, Quijgtcjg, & ^jg (=>>9), jgl

Tcogi, Tcjgi, & Acfg (=>>9).

@roe jhh o` tcf rfvbfwfl works og L^QO òr \Q^, bt wjs lbsaovfrfl tcjt gogf o` sffefl to

tjkf jlvjgtjif o` tcf bgòrejtbog provblfl my aosts o` tcf bglbvblujh flifs suac tcjt tcf

jhiorbtce ajg mf lfsbigfl to sfgsf wcbac flifs wouhl eost hbkfhy hfjl to proebsbgi

posbtbogs, jgl wcbac ogfs wouhl eosthy hfjl to poor posbtbogs. \cjt lbsaovfry proeptfl tcbs

rfsfjrac work.

=.6 Jpphbajtbogs o` Qwjre Bgtfhhbifgaf

Qwjre Bgtfhhbifgaf eftjcfurbstbas cjvf mffg suaafssùhhy jpphbfl to j vfry wblf rjgif o`

`bfhls bgahulbgi aoembgjtorbjh optbebzjtbog promhfes, ùgatbog optbebzjtbog, `bglbgi optbejh

routfs, sacfluhbgi, struaturjh optbebzjtbog, bejif jgjhysbs, ljtj ebgbgi, ejacbgf hfjrgbgi,

mbobgòrejtbas, eflbajh bgòrejtbas, lygjebajh systfes, bglustrbjh promhfes, opfrjtbogs

rfsfjrac, jgl `bgjgaf jgl musbgfss (Jcefl & Ihjsiow, =>:=). Qoef fxjephfs o`

jpphbajtbogs o` swjre bgtfhhbifgaf jrf LGJ sfqufgabgi my cymrblbzjtbog (Mhue, Rjmjr, &

Mhfsj, =>>3), Bejif mougljry fxtrjatbog (Hb, Rujg, Cuj, & @jg, =>:>), flif lftfatbog o`

hjsfr-rjgif bejifs (Yu, Yu, Hfb, Tcjo, & Cujgi, =>:>), ^rflbatbog o` ECA Ahjss BBMbglfrs (Kjrpfgko, Qcb, & Ljb, =>>6), Irbl Dom Qacfluhbgi (Bzjkbjg, Hjljgb, Jmrjcje, &

Qgsfh, =>:>), Brbs rfaoigbtbog (Acfg & Acu, =>>6), Bgtrulfr hoajtbog (Ohbvfrj, Qomrbgco,

Jrjdo, & @jrbjs, =>>2), Cjrlwjrf/Qo`twjrf pjrtbtbogbgi (Jmlfhcjhbe & Cjmbm, =>>2),

protfbg struaturf prflbatbog (]fssoe, ft jh., =>>6), jgl trj``ba bgablfgt lftfatbog (Qrbgbvjsjg,

Hoo, & Acfu, =>>1).


35/128

=6

1

ACJ^\F] 14 FAL^QO JHIO]B\CE

1.: Bgtroluatbog

\cf lbsarftbzjtbog o` tcf ^jrtbahf Qwjep Optbebzjtbog (^QO) tfacgbquf òr tcf \rjvfhhbgi

Qjhfsejg ^romhfe (\Q^) bg tcbs ajsf bs suac tcjt j posbtbog bs j afrtjbg pfreutjtbog o` flifs

tcjt òre j Cjebhtogbjg ayahf. Cfgaf, b` òr fxjephf, wf jrf aogsblfrbgi sbx abtbfs, jhh

bgtfraoggfatfl bg j efsc, tcfg j posbtbog aouhl mf tcf sft o` flifs4 {(:, =), (=, 1), (1,


36/128

=0

Lf`bgbtbog 14 Qumtrjatbog Mftwffg two posbtbog vfators, J jgl M, j sumtrjatbog,

rfprfsfgtfl js J M,bs lf`bgfl js tcf bglfx vjhufs o` flifs bg M, wcbac jrf got bg J. \cf

eotbvjtbog mfcbgl lf`bgbgi sumtrjatbog tcbs wjy jrbsfs `roe tcf two bgstjgafs wcfrf tcbs

opfrjtbog oaaurs bg fqujtbog (:), tcjt bs pmfst S jgl imfst S. \cbs opfrjtbog, bg tcf

ahjssbajh ^QO, bs jbefl jt efjsurbgi tcf lfvbjtbog o` tcf posbtbog o` tcf aurrfgt pjrtbahf `roe

bts pfrsogjh mfst jgl tcf ihomjh mfst, rfspfatbvfhy, so js to ejkf tcf sumsfqufgt posbtbogs to

osabhhjtf jrougl tcosf vjhufs. Gow, sbgaf tcf vfhoabty, P rfprfsfgts tcf guemfr o` flifs to

mf acjgifl bg tcf proafss o` `bglbgi j gfw posbtbog, jgl aogsblfrbgi tcf `jat tcjt tcf rfsuht o`

pmfstSjgl imfstS wbhh bgvjrbjmhy aogtrbmutf to tcbs vjhuf o` P, wf wjgt to ajrry out tcf

sumtrjatbog bg suac j wjy tcjt tcf rfsuht aogstbtutf tcosf flifs bg tcf aurrfgt posbtbog wcbac

jrf got bg tcf pmfst jgl imfst, jgl cfgaf scow us wcbac flifs jrf to mf acjgifl bg tcf

aurrfgt posbtbog to omtjbg tcf gfw ogf. ]fajhh tcjt jt tcbs pobgt, tcfsf vjhufs jrf tcf mfst

kgowg vjhufs, jgl wf jbe to usf tcf bg`orejtbog tcfy aogtjbg to jbl us bg our sfjrac `or gfw

posbtbogs. Jg fxjephf wbhh su``baf to bhhustrjtf tcbs pobgt4

B` 5 {=, 1, 6, 0, 1, ,> >,>,> >,>,> >,>,> >,>,> >,>,>{>.66=:,=, >=,1,>.=1=1,2=16,0, >0,:,=.:::>0,=}

JhhioolgfssBglfx vjhufs o` tcf flifs wbhh mf bgbtbjhbzfl to zfro, jgl wbhh oghy mf upljtfl jt

tcf fgl o` tcf aurrfgt btfrjtbog, tcjt bs j`tfr jhh pjrtbahfs cjvf mffg proafssfl.

jm a


37/128

=9

Js ajg mf sffg `roe tcf fxjephf jmovf, tcf FlifMjgk wbhh jhso mf usfl to trjak tcf

opbgbogs o` tcf lb``frfgt pjrtbahfs jmout j pjrtbauhjr flif. \o lo tcbs, tcf ljtjmjsf wbhh

aogtjbg, òr fjac flif, tcrff bgtfifr vjhufs4 tcf `brst, j, wbhh trjak cow ejgy pjrtbahfs

bgahulfl flifx, tcf sfaogl, m, wbhh trjak cow ejgy pjrtbahfs aogsblfr flif x to mf j iool

flif, js wbhh mf lf`bgfl mfhow, jgl tcf tcbrl, a, wbhh trjak cow ejgy pjrtbahfs aogsblfr flifx

to mf j mjl flif.

(j)\cf aogafpt o` ioolgfss o` jg flif bs lf`bgfl tcus4

B` lurbgi tcf proafssbgi o` j pjrtbahf, tcf `btgfss ùgatbog o` tcf gfw posbtbog

proluafs j mfttfr rfsuht tcjg tcf ogf omtjbgfl jt tcf prfvbous btfrjtbog, tcf

flif bs aogsblfrfl to mf iool. \cjt bs, tcf vjhuf o` m wbhh bgarfjsf my :.

B` og tcf otcfr cjgl, tcf `btgfss ùgatbog o` tcf gfw posbtbog proluafs j worsfrfsuht tcjg tcf ogf omtjbgfl jt tcf prfvbous btfrjtbog, tcf flif bs aogsblfrfl to

mf mjl. \cjt bs, tcf vjhuf o` a wbhh bgarfjsf my :.

B` bt bs gfbtcfr mfttfr gor worsf, gotcbgi wbhh mf logf to tcf flif, or bt bs

aogsblfrfl gfutrjh. \cjt bs j jgl m wbhh rfejbg ugacjgifl.

J`tfr tcf proafssbgi o` fjac pjrtbahf, tcf vjhuf o` j òr jhh flifs aogstbtutbgi tcf gfw

posbtbog wbhh mf bgarfjsfl my :.

(m)

J`tfr jhh pjrtbahfs cjvf mffg proafssfl, tcfioolgfssBglfx

vjhufs o` tcf jhh tcf flifsbg FlifMjgk wbhh mf upljtfl tcus4

:. B` tcf vjhuf o` m bs irfjtfr tcjg or fqujh to cjh` tcf vjhuf o` j, tcf

ioolgfssBglfx o` tcjt flif wbhh mf bgarfjsfl my j aogstjgt, Z. Eorf og tcf

vjhuf o` Z bg sfatbog 1.:.

=. B` tcf vjhuf o` a bs irfjtfr tcjg or fqujh to cjh` tcf vjhuf o` j, tcf

ioolgfssBglfxo` tcjt flif wbhh mf rfluafl my Z.

1. Bg tcf ajsf wcfrf gfbtcfr o` tcf jmovf cjppfgs, or m jgl a jrf tcf sjef, tcf

ioolgfssBglfxo` tcjt flif wbhh got mf acjgifl.

@bgjhhy, tcfrf wbhh mf j `bxfl vjhuf, K,jmovf wcbac b` jgy flifs ioolgfssBglfxajg ift, bt

wbhh mf jhwjys sfhfatfl my pjrtbahfs bg tcfbr ifgfrjtbog o` j gfw posbtbog. \cf jgjhysbs o` Kbs

jhso prfsfgtfl bg sfatbog 1.=.

Lf`bgbtbog 94 \cf upljtf opfrjtbog bg fqujtbog (=) cjs jhso mffg rf-lf`bgfl. Js cjs mffg

fjrhbfr stjtfl, tcf vfhoabty bs tcf guemfr o` flifs to mf rfphjafl bg S. \cfrfòrf, fqujtbog (=)

mfaoefs4

+ 5

+ (1)


38/128

=3

Bg tcf upljtf proafss, + rjgloe flifs wbhh mf sfhfatfl `roe FlifMjgk jgl usfl to

rfphjaf tcf ogfs bg . Bt bgvohvfs motc tcf sfhfatbog o` tcf spfab`ba flifs bg

wcbac jrf

iobgi to mf rfphjafl, jgl sfhfatbgi tcf flifs bg FlifMjgk wcbac wbhh rfphjaf tcfe.

\cf sfhfatbog o` flifs to mf rfphjafl bgwbhh mf bg tcrff stjifs4

Aoeputf tcf jpproxbejtf vjhuf o`

, x, jgl tcfg rjgloehy acoosf, `roe

tcf sft o` bglbafs rfsuhtbgi `roe

,xbglbafs wcosf flifs bgwbhh mf rfphjafl.

Aoeputf tcf jpproxbejtf vjhuf o`

, y, jgl tcfg rjgloehy acoosf, `roe

tcf sft o` bglbafs rfsuhtbgi `roe ,ybglbafs wcosf flifs bg

wbhh mf rfphjafl.

Ycfrf tcf vjhuf o` x + y bs hfss tcjg +, rjgloehy acoosf tcf rfejbgbgi

+(x + y)

flifs wcbac wbhh mf rfphjafl bg .

\cf sfhfatbog o` tcf flifs `roe FlifMjgk wcbac wbhh mf usfl bg tcf rfphjafefgt wbhh jhso mf

tcus4

]jgloehy sfhfat, b` possbmhf, + flifs wcosf ioolgfssBglfx vjhufs jrf irfjtfr tcjg or

fqujh to K. B` tcf guemfr o` flifs sfhfatfl bs got up to + ,tcfg proaffl to =, wbtc z js tcf

guemfr o` flifs jhrfjly sfhfatfl. Fhsf, tfrebgjtf tcf sfhfatbog proafss. \cf jgjhysbs o` Kbs

ibvfg bg sfatbog 1.=.

]jgloehy ifgfrjtf j guemfr, ], wbtcbg tcf bgtfrvjh [>, \U, wcfrf \ bs j vjhuf wcbac wbhh

aogtbguf to vjry tcrouicout tcf btfrjtbvf proafss. \cf vjhuf o` \wbhh mf fhjmorjtfl upog bg

sfatbog 1.1.

@roe FlifMjgk, rjgloehy sfhfat + flifs wcosf ioolgfssBglfxvjhufs jrf cbicfr

tcjg]. Bg tcf ajsf wcfrf tcf guemfr o` flifs wcosfioolgfssBglfx vjhufs jrf cbicfr tcjg]

bs got up to + , tcfg rjgloehy sfhfat tcf rfejbgbgi guemfr o` flifs gfflfl to òre j

Cjebhtogbjg ayahf, fgsurbgi tcjt go flif(s) jrf sfhfatfl wcbac jrf jhrfjly pjrt o` tcf gfwposbtbog. Bt bs wortcy o` gotf tcjt bg sfhfatbgi tcf flifs, tcf sfhfatbog o` tcf hjst flif to òre

tcf gfw posbtbog lofs got rfjhhy cjvf jgy fhfefgt o` stoacjstbabty bg bt, js oghy j pjrtbauhjr

flif bg FlifMjgk ajg mf sfhfatfl tcjt wbhh ejkf up j Cjebhtogbjg ayahf (tcf gfjrfst

gfbicmour). Jhso, bg tcf sfhfatbog o` flifs, ogf eust mf ajrfùh got to sfhfat jg flif tcjt wbhh

cbglfr j Cjebhtogbjg ayahf `roe òrebgi my jlcfrbgi to tcf òhhowbgi ruhfs4

Bg tcf `bgjh hbst o` flifs to aogstbtutf tcf gfw posbtbog, fjac golf eust jppfjr fxjathy twbaf.


39/128

=2

Lo got sfhfat jgy flif wcbac aogsbsts o` j golf wcbac jhrfjly jppfjrs up to two tbefs bg tcf

gfw posbtbog mfbgi ifgfrjtfl.

Lo got sfhfat jgy flif tcjt wbhh ajusf j ayahf to mf òrefl fxafpt bt bs tcf hjst flif, b.f.

prfvfgt tcf òrebgi o` sum-tours.

Lf`bgbtbog 34 @btgfss @ugatbog bs lf`bgfl js tcf sue o` tcf aosts o` jhh tcf flifs tcjt òre

tcf gfw posbtbog.

Lf`bgbtbog 24 \cf vjhuf o` G bs tcf guemfr o` pjrtbahfs bg tcf fgtbrf sfjrac spjaf.

\cfrfòrf, tcf gfw upljtf fqujtbogs òr tcf FA-L^QO jrf tcus4

+ 5

(:)

+ 5 + (1)

1.1

Jgjhysbs o` tcf Aogstjgt, Z

Bt wbhh mf omsfrvfl tcjt tcf vjhuf o` Z wbhh mf mjsfl, to j hjrif fxtfgt, og wcftcfr wf wjgt to

lo eorf fxphorjtbog, or og tcf otcfr cjgl, eorf fxphobtjtbog. Cowfvfr, wf cjvf lfablfl to

sfhfat j vjhuf tcjt wbhh eorf or hfss ejbgtjbg j mjhjgaf mftwffg fxphorjtbog jgl fxphobtjtbog

jgl prfvfgt pjrtbahfs `roe mfaoebgi stuak og hoajh optbeue. B` tcfrf jrf g golfs bg tcf

irjpc, jgl tcfrf jrf eflifs bg FlifMjgk, tcfg tcf sbgiuhjr promjmbhbty o` ogf flif mfbgisfhfatfl, b.f. got aogsblfrbgi tcjt x flifs jrf to mf sfhfatfl jt ogaf to òre j posbtbog, bs

\cfrfòrf, wf aogsblfr bt rfjsogjmhf to sft tcf vjhuf

Z 5 (


40/128

1>

vjhuf, K, jmovf wcbac b` jg flifs ioolgfssBglfx ajg ahbem, bt wbhh jhwjys mf sfhfatfl bs

eotbvjtfl my tcf jttfept to sbgihf out fxafptbogjhhy iool flifs jgl cjstfg aogvfrifgaf

towjrls tcf mfst posbtbog my ejkbgi pjrtbahfs jhwjys sfhfat tcfe, bgstfjl o` wjstbgi tbef

sfhfatbgi flifs wcbac jrf got so iool. Yf jakgowhflif tcjt tcbs bs j efjsurf wcbac ajg

fjsbhy hfjl to tcf pjrtbahfs ifttbgi trjppfl og hoajh optbeue. \cbs bs tcf rfjsog wcy bt wjs

lfablfl tcjt tcf lfabsbog og jg flif mfbgi iool wbhh mf mjsfl og tcf opbgbogs o` jt hfjst

cjh` o` tcf pjrtbahfs wcbac cjvf usfl bt. Eorf so tcf vjhuf o` K wbhh mf sfhfatfl suac tcjt oghy

b` ovfr j sfrbfs o` btfrjtbogs, tcf ioolgfssBglfx (jaueuhjtbvf vjhuf) o` tcjt flif bs up to K

wbhh bt mf pfrejgfgthy sfhfatfl js j iool or lfsbrjmhf flif. Bt scouhl jhso mf gotfl tcjt wcfg

j pjrtbahf bs ejrkbgi jg flif js iool bt bs mjsfl og tcf vjhuf o` tcf `btgfss ùgatbog, wcbac

tjkfs bgto aoigbzjgaf jhh otcfr flifs tcjt ejkf up tcjt posbtbog. \cfrfòrf, otcfr flifs jhso

aogtrbmutf bg tcf lfabsbog, jgl tcf possbmbhbty bs stbhh rfjsogjmhy cbic o` jg flifs

ioolgfssBglfxmfbgi rfluafl mfajusf bt wjs sfhfatfl toiftcfr wbtc otcfr flifs wcosf aosts

jrf cbic, fvfg tcouic bts owg aost bs how. Jhso, wcfrf tcf guemfr o` flifs wcbac cjvf

ioolgfssBglfx vjhufs cbicfr tcjg K bs irfjtfr tcjg tcf vfhoabty, tcf rfqubrfl guemfr o` flifs

wbhh mf rjgloehy sfhfatfl `roe jeogist jhh tcosf wco qujhb`y. Bt bs omvbous tcfg, tcjt tcf

sfhfatbog o` tcf vjhuf o` K cjs to tjkf bgto aogsblfrjtbog tcf promjmbhbty o` jg flif mfbgi

sfhfatfl jeogi tcf g flifs bg tcf flif mjgk, js wfhh js tcf promjmbhbty o` tcjt flif mfbgi

jeogi tcf iroup o` e flifs tcjt wbhh òre j Cjebhtogbjg ayahf wcbac wbhh mf sfhfatfl `roe

jeogi tcf gflifs bg FlifMjgk.

\cf promjmbhbty o` jg flif mfbgi sfhfatfl `roe g flifs bs

. \cfrfòrf, tcf promjmbhbty o`

sfhfatbgi jg flif jeogi tcf e flifs to mf sfhfatfl `roe tcf gflifs bg FlifMjgk bs

.

\cfrfòrf, tcf vjhuf o` K bs fxprfssfl js

K 5

(6)

1.6 Jgjhysbs o` tcf Pjhuf, \

\cf vjhuf \ bs bgtroluafl so js to ibvf tcf flifs wcosf ioolgfssBglfx vjhufs jrf cbicfr tcjg

tcosf o` otcfrs j ebgutfhy mfttfr acjgaf o` mfbgi sfhfatfl jt fjac suaafssbvf btfrjtbog. Bt bs

beportjgt tcjt tcf vjhuf o` \ stjrt jt j sejhh vjhuf jgl proaffl cbicfr js tcf ejxbeue

ioolgfssBglfx vjhuf o` tcf flifs bgarfjsfs, so js to aogtbgujhhy ibvf mfttfr flifs j eorf


41/128

1:

prospfat tcjg poorfr ogfs. Qbgaf tcf vjhuf o` \ bs lygjeba, wf wbhh `brst o` jhh lf`bgf tcf

bgbtbjh vjhuf o` \ js

5

(0)

\cf sumsfqufgt vjhufs o` \ wbhh mf lf`bgfl js

5 ejx{, , , } (9)

Ycfrf4

pb5 tcf ioolgfssBglfx o` flif b bg FlifMjgk

g 5 guemfr o` flifs bg FlifMjgk

k 5 tcf guemfr o` btfrjtbog

\cus, \wbhh jhwjys ejbgtjbg j hfjl jcfjl o` tcf cbicfst possbmhf ioolgfssBglfx vjhuf jg

flif ajg cjvf.

1.0 Qbeuhjtbog

J scort sbeuhjtbog o` tcbs jhiorbtce wbhh mf logf so js to scow tcf gjturf o` tcf jhiorbtce

jgl ibvf j ahfjrfr pbaturf o` cow bt opfrjtfs. Bg tcbs sbeuhjtbog o` tcf FA-L^QO jhiorbtce,

tcf sfjrac spjaf wbhh mf ejlf o` = pjrtbahfs iobgi tcrouic tcrff btfrjtbogs. \cf sfhfatfl

pjrjeftfrs jrf `or bhhustrjtbog purposf oghy. Mjsfl og rfvbfwfl hbtfrjturf, pjrjeftfrs wbtc

mfttfr rfsuhts ajg jgl wbhh mf sfhfatfl `or tcf bephfefgtjtbog o` tcf jhiorbtce. J 0-abty \Q^

wbhh mf aogsblfrfl, wbtc tcf irjpc mfhow4

a: 5 >.3, a=5 >.3, w 5 :

: =

1


42/128

1=

Bgbtbjhbzjtbog

^jrtbahf :4

5 {(:,0), (0,=), (=,1), (1, >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> 5 {>:,=, >=,1, >1,6,0, >0,:, >0,=, >0,1, >0,,>,> >,>,> >,>,> >,>,>>=,6, >6,:, >6,1,>1,:}

Irjpc scowbgi 4

@roe (.>09, \>5

5 >.>09,K 5

5 >.


43/128

11

:stBtfrjtbog, wcfg t bg (6) bs :

^jrtbahf :4

Hft 5 >.=, 5 >.3

5 {=, :9

Flifs to mf rfphjafl4 :, =, 1, ,> >,>,> >,>,> >,>,> >,>,> :,:,> :,:,> >,>,> :,:,> 5 {>:,=, >=,1, >1,6,0, >0,:, >0,=, >0,1, >0,,>,> :,:,> :,:,> >,>,>>=,6, >6,:, >6,1,>1,:}

Gotf tcjt tcf vjhufs o` jjgl mcjvf mffg upljtfl mjsfl og lf`bgbtbog 0.

^jrtbahf =

Hft 5 >.6, 5 >.0

5 {=,1,6,0}

5 =

5 {=,1,0}

5 :

5

arimfst S

5 : * = + = + :

5 6

Hft ] 5 >.>


44/128

1 >,>,> :,:,> >,>,> >,>,> =,=,> =,=,> >,>,> =,=,> 5 {>:,=, >=,1, >1,6,0, >0,:, >0,=, >0,1, >0,,>,> =,=,> :,:,> :,:,>>=,6, >6,:, >6,1,>1,:}

Aoeeugbajtbog

5 {(:,=), (=,,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> 5 {>.>09:,=,>.>09=,1, >1,.>096,0, >0,:, >0,=,>.>090,1,

>,>,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,>>.>090,=,6,>.>096,:,>.>096,1,>.>091,:}

Irjpc scowbgi 4

@roe (2),\ 5 >.:1.1, 5 >.

5 {} 5 >

: =

1

+ >

5 0

5 0

Hft ] 5 >.>=6

Flifs to mf rfphjafl4 :, =, 1, ,>,> >,>,> :,>,: >,>,> >,>,> >,>,> 5 {>.>09:,=,>.>09=,1, >1,.>096,0, >0,:, >0,=,>.>090,1,

:,>,: >,>,> :,>,: >,>,> :,>,: >,>,> :,>,:>.>090,=,6,>.>096,:,>.>096,1,>.>091,:}

^jrtbahf =

Hft 5 >.2, 5 >.=

5 {=, 6} 5 =

5 {:,

5 arimfst S

5 :*6 + = + >

5 9

5 0(ahjepbgi o` vfhoabty)

Hft ] 5 >.:>6

Flifs to mf rfphjafl4 :, =, 1,


46/128

10

:,:,> :,>,: :,:,> >,>,> :,:,: >,>,> >,>,> >,>,> 5 {>.>09:,=,>.>09=,1, >1,.>096,0, >0,:, >0,=,>.>090,1,

:,>,: :,:,> :,>,: :,:,> :,>,: >,>,> :,>,:>.>090,=,6,>.>096,:,>.>096,1,>.>091,:}

Aoeeugbajtbog

5 {(:,=), (=,6), (6,0), (0,1), (1,,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,> 5 {>.:1=,1,>.>091,.>096,0, >0,:, >0,=,>.>090,1,

>,>,> >,>,> >,>,> >,>,> >,>,> >,>,> >,>,>>0,.>09.>09=,6, >6,:,>.>096,1,>1,:}

\ 5 >.=>:, `roe (2).

Jt tcf fgl o` tcf sfaogl btfrjtbog, tcf `btgfss vjhuf omtjbgfl ajg mf sffg to mf jpprojacbgi tcf fxjat

sohutbog, jgl wbhh fvfgtujhhy aogvfrif to j sohutbog wcbac bs vfry ahosf to bt. Bt bs jhso bgtfrfstbgi to

omsfrvf cow soef o` tcf how-aost flifs jhrfjly cjvf cbicfr ioolgfssBglfx vjhufs tcjg otcfrs. \cf

fxjat sohutbog bs {(:,=), (=,6), (6,1), (1,0), (0,


47/128

19

1.9 FA-L^QO @orejh Jhiorbtce

Mfhow bs j `orejh prfsfgtjtbog o` tcf FA-L^QO jhiorbtce4

Jhiorbtce BB4 FA L^QO Jhiorbtce

:4 ]jgloehy bgbtbjhbzf tcf wcohf swjre

=4 Aoeputf Z jgl K, jgl \> // usbgi fqujtbogs (

jgl jhh ioolgfssBglfx vjhufs to >.>

4 upljtf FlifMjgk // usbgi lf`bgbtbog 0(j)

::4 }

:=4 `or(t 5 :; t85swjresbzf; t++) {

:14 b` (`(pmfstt)8 `(imfst)) imfst 5 pmfstt

:< }

:64 upljtf FlifMjgk // usbgi lf`bgbtbog 0(m)

:04 aoeputf \k // usbgi fqujtbog (9):94 }

: =

1


48/128

13

>ICz wbtc IM ]JE og Ybglows 9. \cf vjhufs o` a:jgl a=

wfrf motc sft jt :.) tbefs wbtc ljtj `or < to =>-

abty \Q^s (aoephftf irjpcs), jgl tcf rfsuhts o` tcf tcrff vjrbjgts wfrf aoepjrfl mjsfl og tcf

`ohhowbgi arbtfrbj4

Jvfrjif guemfr o` btfrjtbogs

Jvfrjif tbef tjkfg bg sfaogls

Efjg Lfvbjtbog

Ebgbeue Lfvbjtbog4 \cbs arbtfrbog wjs bgtroluafl so js to ibvf j ahfjr pbaturf o` tcf

qujhbty o` tcf rfsuht, mfjrbgi bg ebgl tcf possbmbhbty tcjt fvfg wbtc j rfjsogjmhy how

efjg lfvbjtbog, tcf sohutbogs aouhl mf ahustfrfl jrougl j vfry poor rfibog. Ebgbeue

Lfvbjtbog cfrf bs ibvfg js tcf jvfrjif lb``frfgaf mftwffg fjac sohutbog jgl tcf mfst

sohutbog omtjbgfl ovfr tcf 6> btfrjtbogs

Guemfr o` mfsts4 \cf guemfr o` tbefs tcf mfst vjhuf oaaurs ovfr tcf 6> btfrjtbogs

Mfst Ebgbeue Aost4 \cf mfst `btgfss vjhuf omtjbgfl ovfr tcf 6> rugs.

\cf tfrebgjtbog arbtfrbog wjs sft jt aogvfrifgaf, b.f. wcfg g * :>aogsfautbvf btfrjtbogs ibvf

tcf sjef imfst, wcfrf g bs tcf guemfr o` abtbfs.

\cf irjpcs mfhow scow tcf omsfrvfl trfgls4


49/128

12

@roe tcf irjpc, bt ajg mf sffg tcjt tcf jvfrjif guemfr o` btfrjtbogs bgarfjsfs stfjlbhy wbtc

bgarfjsf bg guemfr o` abtbfs òr jhh tcrff vjrbjgts. Cowfvfr, bt wbhh mf gotbafl tcjt `roe tcf

:=-abty pobgt, tcf mjsba FAL^QO vjrbjgt bgarfjsfs jt j showfr rjtf tcjg tcf otcfr vjrbjgts.

\cbs efjgs tcjt bt iofs tcrouic `fwfr btfrjtbogs tcjg tcf otcfrs wbtc bgarfjsf bg abty guemfr.

\cbs cowfvfr, lofs got gfafssjrbhy efjg tcjt bt pfròres mfttfr. Og ogf cjgl, bt aouhl efjg

tcjt bt bs jmhf to jacbfvf tcf mfst sohutbog wbtc `fwfr btfrjtbogs, jgl og tcf otcfr cjgl, tcjt bt

ifts stuak og hoajh optbej. Bt wbhh jhso mf omsfrvfl tcjt og tcf sejhhfr abty bgstjgafs (),tcf FAL^QO jgl FAL^QO(\) vjrbjgts cjl howfr vjhufs tcjg tcf FAL^QO(M) vjrbjgt.

\cf trfgl cfrf bs sbebhjr to tcf ogf omsfrvfl bg tcf irjpc òr jvfrjif guemfr o` btfrjtbogs,

jgl tcf aogahusbogs lrjwg `roe tcf ogf jmovf jhso jpphy to bt. Bg jllbtbog, wf omsfrvf tcjttcf FAL^QO vjrbjgt tjkfs shbicthy hogifr tbef tcjg tcf FAL^QO(\) vjrbjgt.

>

:>>

=>>

1>>

>

6>>

0>>

9>>

AB\BFQ

:=

AB\BFQ

:

AB\BFQ

JvfrjifGuemfro`Btfrj

tbogs

Jvfrjif Guemfr o` Btfrjtbogs

FAL^QO(M)

FAL^QO(\)

FAL^QO

>

=

:=

: btfrjtbogs bs vfry how.

\cfrfòrf tcf rfsuhts mfaoef eorf jgl eorf ugrfhbjmhf js tcf guemfr o` abtbfs bgarfjsfs.

\cbs irjpc ibvfs j vfry ahfjr pbaturf o` tcf pfròrejgaf o` tcf tcrff vjrbjgts. \cf FA^LO

vjrbjgt pfròres shbicthy mfttfr tcjg tcf FAL^QO(\)vjrbjgt, wcbhf tcf FAL^QO(M) vjrbjgt

cjs j vfry poor pfròrejgaf aoepjrfl to tcfe. \cbs rf-fgòrafs tcf aogahusbog tcjt tcf

rfsuhts ibvfg my tcf FAL^QO(M) vjrbjgt jrf cbichy ugrfhbjmhf, fspfabjhhy wbtc cbic abty

guemfrs.

>

=

:=

:

AB\BFQ

:=

AB\BFQ

:

AB\BFQ

EfjgLfvbjtbog

Efjg Lfvbjtbog

FAL^QO(M)

FAL^QO(\)

FAL^QO

>

:>

=>

1>

6>

0>

9>

3>

2>

AB\BFQ

:=

AB\BFQ

:

AB\BFQ

EbgbeueL

fvbjtbog

Ebgbeue Lfvbjtbog

FAL^QO(M)

FAL^QO(\)

FAL^QO


51/128

rugs.

\cfrf bs jg jirffefgt bg rfsuht jeogi jhh tcrff vjrbjgts, ugtbh tcf :=-abty \Q^, wcfrf tcf

FAL^QO(M) vjrbjgt vffrs o`` bgto cbicfr, jgl o` aoursf hfss-lfsbrjmhf vjhufs.

@roe tcf omsfrvjtbogs ejlf `roe tcf 0 irjpcs jmovf, bt bs fvblfgt tcjt tcf FAL^QO(M)

vjrbjgt bs tcf poorfst, jgl tcjt bts sejhhfr fxfautbog tbefs jgl guemfr o` btfrjtbogs jrf js j

rfsuht o` tcf `jat tcjt bt fjsbhy ifts stuak og hoajh optbej. @urtcfreorf, tcf FAL^QO vjrbjgt

>

:>

=>

1>

6>

0>

AB\BFQ

:=

AB\BFQ

:

AB\BFQ

Guemfro`Mfsts

Guemfr o` Mfsts

FAL^QO(M)

FAL^QO(\)

FAL^QO

>

6>

:>>

:6>

=>>

=6>

1>>

< AB\BFQ0 AB\BFQ3 AB\BFQ :>

AB\BFQ

:=

AB\BFQ

:

AB\BFQ

MfstEbgbeueA

ost

Mfst Ebgbeue Aost

FAL^QO(M)

FAL^QO(\)

FAL^QO


52/128


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57/128


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59/128

>% jaaurjtf. Cowfvfr, wcfg

tcbs bg`orejtbog bs bgtfirjtfl bgto tcf ^QO jhiorbtce, bts usf`uhgfss jgl f``babfgay mfaoefs

fvblfgt.

Og tcf ]fsuht pjgf, tcf aurrfgt posbtbog mfbgi fxphorfl bs wcjt bs scowg.

Jt tcf fgl o` tcf fxfautbog, tcf ]fsuht pjgf scows tcf mfst tour, jgl mfhow bt, eorf ljtj

jmout tcf fxfautbog4


60/128

6>

Jt tcf fgl o` tcf fxfautbog, tcf usfr ajg sjvf tcf rfsuht my ahbakbgi og tcf @bhf-7Qjvf Js

efgu, wcfrf cf ajg sjvf cbs rfsuhts4


61/128

6:


62/128

6=

}

pumhbaFlif(Qtrbgi `roe, Qtrbgi to, loumhfaost) {usf@rfqufgay5 >;iool5 >;mjl5 >;

tcbs.`roe5 `roe;tcbs.to5 to;tcbs.aost5 aost;promjmbhbty5 >.>;

}

/*** Eftcol to ift j aopy o` tcf ajhhbgi flif** Nrfturgj aopy o` tcbs flif*/

NOvfrrblfpumhbaFlif ahogf() {

Flif lueey 5 gfwFlif();lueey.sft@roe(`roe);lueey.sft\o(to);lueey.sftMjl(mjl);lueey.sftAost(aost);lueey.sftIool(iool);lueey.sft^romjmbhbty(promjmbhbty);lueey.sftVsf@rfqufgay(usf@rfqufgay);rfturglueey;

}

/**

* Lofs trjgsbtbvf aoepjrbsog o` flifs `or syeeftrba irjpcs to sff b` tcfy* jrf fqujh** Npjrjeaoepjrfl\o* tcf sfaogl flif mfbgi aoepjrfl to* Nrfturgb` tcf motc flifs mfbgi aoepjrfl jrf fqujh*/pumhbamoohfjgfqujhs(Flif aoepjrfl\o) {

rfturg((`roe.fqujhs(aoepjrfl\o.ift@roe()) && to.fqujhs(aoepjrfl\o.ift\o())) || (to.fqujhs(aoepjrfl\o.ift@roe()) &&

aoepjrfl\o.ift\o().fqujhs(`roe)));

}

/*** Nrfturgmjl tcf 'mjlgfss' bglfx o` tcbs Flif*/pumhbabgtiftMjl() {

rfturgmjl;}

/*** Nrfturgtcf aost o` tcbs Flif*/pumhbaloumhfiftAost() {

rfturgaost;

}


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61

/*** Nrfturg8m7{Naolf `roe4} 8/m7tcf souraf vfrtfx o` tcbs Flif*/pumhbaQtrbgi ift@roe() {

rfturg`roe;}

/*** Nrfturgiool tcf 'ioolgfss' bglfx o` tcbs Flif*/pumhbabgtiftIool() {

rfturgiool;}

/**** Nrfturgtcf promjmbhbty jssoabjtfl wbtc tcbs flif*/pumhbaloumhfift^romjmbhbty() {

rfturgpromjmbhbty;}

/*** Nrfturgto tcf lfstbgjtbog vfrtfx o` tcbs flif*/pumhbaQtrbgi ift\o() {

rfturgto;}

/*** Nrfturgusf@rfqufgay tcf guemfr o` pjrtbahfs tcjt cjvf usfl tcbs flif

* wbtcbg ogf btfrjtbog*/pumhbabgtiftVsf@rfqufgay() {

rfturgusf@rfqufgay;}

/*** Npjrjemjl* tcf 'mjlgfss' bglfx o` tcbs Flif*/pumhbavoblsftMjl(bgtmjl) {

tcbs.mjl5 mjl;}

/*** Npjrjeaost* tcf aost o` tcbs Flif*/pumhbavoblsftAost(loumhfaost) {

tcbs.aost5 aost;}

/*** Npjrje`roe* tcf souraf vfrtfx o` tcbs Flif*/

pumhbavoblsft@roe(Qtrbgi `roe) {tcbs.`roe5 `roe;


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6;fhsf

tcbs.promjmbhbty5 promjmbhbty;}

/*** Npjrjeto* sft tcf lfstbgjtbog vfrtfx o` tcbs Flif*/pumhbavoblsft\o(Qtrbgi to) {

tcbs.to5 to;}

/**

* Npjrjeusf@rfqufgay* sft tcf usf `rfqufgay o` tcbs Flif*/pumhbavoblsftVsf@rfqufgay(bgtusf@rfqufgay) {

tcbs.usf@rfqufgay5 usf@rfqufgay;}

NOvfrrblfpumhbaQtrbgi toQtrbgi() {

rfturg"("+ `roe+ ", "+ to+ ")";}

}


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66

beportdjvj.utbh.Btfrjtor;beportdjvj.utbh.Hbst;

beportfxafptbog.FlifQftFxafptbog;beportfxafptbog.BgaogsbstfgtQtjtfFxafptbog;beportfxafptbog.BtfrjmhfFlifHbst;

beportfxafptbog.QbzfFxafflflFxafptbog;

pumhbaahjssFlifQft bephfefgtsBtfrjtor8Flif7, Btfrjmhf8Flif7 {prbvjtfHbst8Flif7 flifHbst;prbvjtf`bgjhHbst8Flif7 bgaoephftfHbst;prbvjtf`bgjhbgttypf;pumhbastjtba`bgjhbgt^J]\BAHF5 >;pumhbastjtba`bgjhbgtFLIFMJGK5 :;pumhbastjtba`bgjhbgtEJS_PJHVF5 :>>>>>>;

pumhbastjtbamoohfjgbsCjebhtogbjgAyahf(Hbst8Flif7 flifs, Flif lue) {Flif lueey 5 lue.ahogf();BtfrjmhfFlifHbst lueeyHbst 5 gfwBtfrjmhfFlifHbst(flifs);

Qtrbgi lueeyQtrbgi 5 "";Qtrbgi lfstbgjtbog 5 "";Flif gfxtFlif 5 lueeyHbst.gfxt();bgtaougt 5 >;moohfjgòugl 5 `jhsf;wcbhf(lueeyHbst.cjsGfxt()) {

b`(!òugl) {b`(aougt 7 lueeyHbst.sbzf())

mrfjk;b`(lueey.ift@roe().fqujhs(gfxtFlif.ift@roe())) {

lueeyQtrbgi 5 gfxtFlif.ift\o();lfstbgjtbog 5 lueey.ift\o();

lueeyHbst.rfeovf();òugl 5 truf;aougt 5 >;

} fhsfb`(lueey.ift@roe().fqujhs(gfxtFlif.ift\o())) {lueeyQtrbgi 5 gfxtFlif.ift@roe();lfstbgjtbog 5 lueey.ift\o();lueeyHbst.rfeovf();òugl 5 truf;aougt 5 >;

} fhsfb`(lueey.ift\o().fqujhs(gfxtFlif.ift@roe())) {lueeyQtrbgi 5 gfxtFlif.ift\o();lfstbgjtbog 5 lueey.ift@roe();lueeyHbst.rfeovf();

òugl 5 truf;aougt 5 >;

} fhsfb`(lueey.ift\o().fqujhs(gfxtFlif.ift\o())) {lueeyQtrbgi 5 gfxtFlif.ift@roe();lfstbgjtbog 5 lueey.ift@roe();lueeyHbst.rfeovf();òugl 5 truf;aougt 5 >;

}gfxtFlif 5 lueeyHbst.gfxt();aougt++;

} fhsf{b`(aougt 7 lueeyHbst.sbzf())

mrfjk;b`(gfxtFlif.ift@roe().fqujhs(lueeyQtrbgi)) {


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60

lueeyQtrbgi 5 gfxtFlif.ift\o();lueeyHbst.rfeovf();aougt 5 >;

} fhsfb`(gfxtFlif.ift\o().fqujhs(lueeyQtrbgi)) {lueeyQtrbgi 5 gfxtFlif.ift@roe();lueeyHbst.rfeovf();

aougt 5 >;}b`(lueeyQtrbgi.fqujhs(lfstbgjtbog))

rfturgtruf;b`(!lueeyHbst.cjsGfxt())

mrfjk;gfxtFlif 5 lueeyHbst.gfxt();aougt++;

}}rfturg`jhsf;

}

prbvjtfbgtfxpfatflQbzf;

/*** Nrfturg{Naolf bgt} tcf fxpfatflQbzf*/pumhbabgtiftFxpfatflQbzf() {

rfturgfxpfatflQbzf;}

prbvjtfbgtbglfx;prbvjtfloumhfjvfrjifAost;

/*** Nrfturg{Naolf loumhf} tcf jvfrjifAost*/pumhbaloumhfiftJvfrjifAost() {

rfturgjvfrjifAost;}

/*** Npjrje8m7jvfrjifAost8/m7tcf {Naolf loumhf} jvfrjifAost to sft*/pumhbavoblsftJvfrjifAost(loumhfjvfrjifAost) {

tcbs.jvfrjifAost5 jvfrjifAost;}

/*** FlifQft Aogstruator.** Npjrje{Naolf bgt} 8m7fxpfatflQbzf48/m7\cf fxpfatfl sbzf o` tcf FlifQft* Npjrje{Naolf bgt} 8m7typf48/m7\cf typf o` FlifQft, FLIFMJGK or

^J]\BAHF*/pumhbaFlifQft(bgtfxpfatflQbzf, bgttypf) {

bgaoephftfHbst5 gfwJrrjyHbst8Flif7();b`(fxpfatflQbzf 55 EJS_PJHVF)

flifHbst5 gfwJrrjyHbst8Flif7();tcbs.fxpfatflQbzf5 fxpfatflQbzf;

tcbs.typf5 typf;bglfx5 >;


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69

}

pumhbaFlifQft() {tcbs(EJS_PJHVF, FLIFMJGK);

}

/*** Eftcol to jll jg flif to tcbs FlifQft, prfvfgtbgi luphbajtf flifs, jgl

bg* tcf ajsf o` 8mr7* j ^J]\BAHF, eorf tcjg = golfs mfbgi bg tcf flifQft, js wfhh js prfejturf* Cjebhtogbjg ayahfs.** Npjrjef* \cf flif to mf jllfl.* Nrfturg{Naolf `jhsf}, b` tcf jllbtbog bs got suaafss`uh, jgl* {Naolf truf} b` bt bs.* NtcrowsQbzfFxafflflFxafptbog* b` tcf jttfeptfl jllbtbog wbhh rfsuht bg tcf spfab`bfl

* fxpfatflQbzf mfbgi fxafflfl*/pumhbamoohfjgjll(Flif f) tcrowsFlifQftFxafptbog {

bgt`roe 5 >;bgtto 5 >;`or(Flif f: 4 bgaoephftfHbst) {

b`(f:.fqujhs(f))rfturg`jhsf;

b`(f.ift@roe().fqujhs(f:.ift@roe())) {`roe++;

}b`(f.ift@roe().fqujhs(f:.ift\o())) {

`roe++;}b`(f.ift\o().fqujhs(f:.ift@roe())) {

to++;}b`(f.ift\o().fqujhs(f:.ift\o())) {

to++;}

}

b`((`roe 75 = || to 75 =) && typf55 ^J]\BAHF)rfturg`jhsf;

b`(bgaoephftfHbst.sbzf() 75 fxpfatflQbzf)

tcrowgfwQbzfFxafflflFxafptbog("\cf spfab`bfl fxpfatfl sbzf cjs mffg fxafflfl!");

b`(bgaoephftfHbst.sbzf() 8 fxpfatflQbzf- :&& bgaoephftfHbst.sbzf() 7 : && typf!5 FLIFMJGK) {b`(bsCjebhtogbj

bsc thesis - an enhanced communication discrete particle swarm optimization algorithm for the...

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