bsc thesis - an enhanced communication discrete particle swarm optimization algorithm for the...
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7/25/2019 BSc Thesis - An Enhanced Communication Discrete Particle Swarm Optimization Algorithm for the Travelling Sales
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JG FGCJGAFL AOEEVGBAJ\BOG LBQA]F\F^J]\BAHF QYJ]E O^\BEBTJ\BOG CFV]BQ\BA@O] \CF \]JPFHHBGI QJHFQEJG ^]OMHFE
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^O^OOHJ, ^F\F] JROKVGHF
VD/=>>3/GQ/>1=>
BG ^J]\BJH @VH@BHEFG\ O@ \CF ]FZVB]FEFG\@O] \CF LFI]FF O@ M.Qa BG AOE^V\F] QABFGAF,
LF^J]\EFG\ O@ AOE^V\F] QABFGAF
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EJR, =>:
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LFAHJ]J\BOG
B, ^O^OOHJ ^F\F] JROKVGHF wbtc tcf ejtrbauhjtbog guemfr VD/=>>3/GQ/>1=>,
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.
^O^OOHJ ^F\F] JROKVGHF Ljtf
(VD/=>>3/GQ/>1=>)
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J^^]OPJH
\cbs bs to afrtb`y tcjt tcbs rfport wjs wrbttfg my ^O^OOHJ ^F\F] JROKVGHF o` tcf
ejtrbauhjtbog guemfr VD/=>>3/GQ/>1=>. \cf work cjs mffg rfjl jgl jpprovfl to cjvf efttcf rfqubrfefgts o` tcf Lfpjrtefgt o` Aoeputfr Qabfgaf, @jauhty o` Gjturjh Qabfgaf,
Vgbvfrsbty o` Dos, ^hjtfju Qtjtf, Gbifrbj.
___________________________ __________________
Lr. G. P. Mhjejc Ljtf
(Qupfrvbsor)
___________________________ _____________________
Lr. \jgko Bscjyj Ljtf
(Cfjl o` Lfpjrtefgt)
___________________________ _____________________
Lr. M. K. Jhfsf Ljtf
(Fxtfrgjh Fxjebgfr)
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bv
LFLBAJ\BOG
\cbs prodfat bs lflbajtfl to Iol tcf @jtcfr, tcf Qog jgl tcf Cohy Qpbrbt, jgl to eyhovbgi pjrfgts, ^jstor jgl Ers ^opoohj.
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JAKGOYHFLIFEFG\Q
Ey pro`ougl irjtbtulf iofs to Jhebicty Iol, tcf @jtcfr, Qog jgl Cohy Qpbrbt, `or cbs
ebicty Cjgl upog ef rbict `roe ey mbrtc tbhh toljy. Cf cjs iublfl ey stfps jhh tcf wjy
tcrouic tcbs uglfrirjlujtf lfirff. O` j trutc, cf cjs mffg tcf Ogf wco cjs pfr`fatfl jhh
tcbgis `or ef. B je gotcbgi wbtcout Cbe. Ybtc jhh ey cfjrt, B sjy, \cjgk you @jtcfr.
Ey sbgafrf jpprfabjtbog iofs to ey pjrfgts, ^jstor & Ers. ^opoohj, Q. @. `or 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 `or tcf prbvbhfif o` aoebgi bgto tcbs worhl tcrouic you. B jhso wjgt to jpprfabjtf
ey sbmhbgis, ^fjaf, ^rjbsf jgl ^juh `or tcfbr aogstjgt support jgl fgaourjifefgt. Rou cjvf
mffg ey `rbfgls jgl pjrtgfrs bg proirfss, jgl j mhfssbgi to ef. B tcjgk Iol `or 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 `or 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.
Yortcy o` spfabjh efgtbog cfrf bs ey irfjt jgl jebjmhf `rbfgl, Mrugo Bsojrlb. Rou
jrf j Iol-sfgl. Rou cfhpfl ef so euac, out o` ejgy muiiy sbtujtbogs, jgl wf cjl so euac
`ug 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 `or jhh tcf blfjs
wfvf scjrfl jgl jhh tcf `ug wfvf cjl. Bts jhwjys j phfjsurf to tjhk wbtc you. Got to `orift
ey irfjt `rbfgls bg VD^A, Ejmjs, Dulf, Feejgufh, Omb, Fglff, Fefkj, Gjtcjgbfh, Gjsbru,
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vb
^rfabous, Iforif, Jkos jgl jhh tcf otcfr irfjt proirjeefrs. My Iols irjaf, you wbhh io
phjafs.
Ey jpprfabjtbog jhso iofs to ey jhh ey woglfr`uh hfaturfrs, fspfabjhhy Er. B`fjgyb
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 woglfr`uh 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 woglfr`uh pjstor, ^jstor Ohuimfgij Oiug`owokjg. Rour bepjat bg
ey hb`f wbhh gfvfr mf `oriottfg.
@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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vbb
\JMHF O@ AOG\FG\Q
LFAHJ]J\BOG...................................................................................................................... bb
J^^]OPJH ........................................................................................................................... bbb
LFLBAJ\BOG......................................................................................................................... bv
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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vbbb
=.=.9 Qohvbgi Cjrl ^romhfes ............................................................................ :
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7/25/2019 BSc Thesis - An Enhanced Communication Discrete Particle Swarm Optimization Algorithm for the Travelling Sales
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bx
ACJ^\F] @BPF4 AOGAHVQBOG JGL ]FAOEEFGLJ\BOG ................................... 99
6.: Aogahusbog ...................................................................................................... 99
6.= ]faoeefgljtbog ............................................................................................ 99
]f`frfgafs ............................................................................................................................... 93
Jppfglbx B .............................................................................................................................. 36
Jppfglbx BB ............................................................................................................................. 39
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x
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 aoe`ort. 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, `ugljefgtjh
aogafpts, suac js jhiorbtce, or aoeputjtbogjh promhfe, cjl mffg `orejhbzfl dust soef
yfjrs mf`orf (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 `ore o` tcf \Q^
aouhl cjvf mffg `brst stulbfl bg :21> my Kjrh Efgifr (Aook, =>>9). Eftcols `or 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 `or
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 `or 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 `orejh 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 `or 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 `orefl 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 `or 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 `ugatbog wcbac lftfrebgfs tcf lfsbrjmbhbty o` j pjrtbauhjr
sohutbog.
:.= Jbe jgl Omdfatbvfs
\cbs prodfat jbes to `oreuhjtf jg jhiorbtce `or 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 `ohhowbgi 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 `oaus og ^jrtbahf Qwjep Optbebzjtbog (cfgaf`ortc ^QO)
1. Vsf tcf ahjssbajh \rjvfhhbgi Qjhfsejg ^romhfe (cfgaf`ortc \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 js`ohhows4 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
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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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\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) `or 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 `oausfs og bgtrjatjmhf ogfs.
Bgtrjatjmhf ^romhfes
\cfsf jrf promhfes `or wcbac go sohvbgi jhiorbtce bs kgowg. Bt cjs mffg scowg tcjt `or tcfsf
promhfes, tcf howfr mougl og tcf tbef-lfejgl o` jgy possbmhf jhiorbtce to sohvf tcfg bs j
`ugatbog 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
`orejhhy lf`bgf bgtrjatjmhf promhfes tcfrf`orf 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. \cfrf`orf, tcf jbe o` rfsfjracfrs
cjs mffg to mypjss fxcjustbvf sfjrac jgl, usbgi ahufs `roe tcf bgput, irfjthy gjrrow lowg
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tcf sfjrac spjaf to jg fxtfgt wcfrf j aorrfat sohutbog ajg mf `ougl 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` `or 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 `or 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 `or Gog-lftfrebgbstba ^ohygoebjh 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``orl, =>>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.
^ohygoebjh-tbef ]fluatbog
Mf`orf j `orejh 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
`ugatbog`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 `ugatbog 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 `or promhfe M to mf fxfautfl
provblfs j efacjgbse wcfrfmy tcf output omtjbgfl my tcbs jhiorbtce `or promhfe M
ajg mf trjgshjtfl mjak bgto jg output `or promhfe J.
\cf jhiorbtce `or 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 `or M, wf ajg sjy tcjt, bg tfres o` bts `ugljefgtjh 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` `or 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
`or 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 `oreuhj bg AG@ jsks b` tcfrf bs j possbmhf
jssbigefgt wcbac ejkfs tcf moohfjg `oreuhj truf. \cjt bs, tcf bgput to j QJ\ bs j moohfjg
`oreuhj hbkf
,
jgl tcf qufstbog bs wcftcfr bt bs possbmhf to jssbig Moohfjg vjhufs to tcf vjrbjmhfs x, y, z,
so tcjt tcf `oreuhj 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 `or eftcols `or 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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:1
\cf stuly o` tcf \Q^ cjs iogf mfyogl dust tcf ahjssbajh promhfe, bt mfbgi usfl js j
`ougljtbog `or 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 `ore 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 `or 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 `orsabfgtbsts 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 `ores j mjsbs `or
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 `or 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 trjgs`orebgi bt bgto j `ore wcbac bs sohvjmhf bg rfjsogjmhf
tbef.
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:6
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
`orejh lf`bgbtbog o` tcf aogafpt. Jaaorlbgi to tcfe, tcf aogafpt cjs, mjsfl og tcf vjrbous
lf`bgbtbogs pro``frfl, mffg jssoabjtfl wbtc tcf `ohhowbgi lbefgsbogs o` efjgbgi4
ugafrtjbgty o` outaoef, mjsbs bg bgaoephftf kgowhflif, beprovfefgt o` pfr`orejgaf jgliubljgaf o` lfabsbog ejkbgi. J`tfr rfvbfwbgi vjrbous lf`bgbtbogs, tcfy ijvf j `orejh
lf`bgbtbog wcbac cjreogbzfs tcfsf `our 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 usf`uh bg provblbgi j
prjatbajh sohutbog, mut wcbac ogf cjs rfjsog to mfhbfvf wbhh mf usf`uh, jgl wcbac[ejy mfU jllfl to j promhfe sohvbgi systfe bg fxpfatjtbog tcjt og jvfrjif tcf
pfr`orejgaf 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 `or 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 `ohhowbgi 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 & ^otvbg, =>:>) jgl Btfrjtfl Hoajh Qfjrac
(Hourfgo, Ejrtbg, & Qttzhf, =>:>).
^opuhjtbog-mjsfl vs. sbgihf pobgt sfjrac4 ^opuhjtbog-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 `oaus o`
tcbs rfsfjrac, bs j popuhjtbog-mjsfl, gjturf-bgspbrfl efeory usbgi eftjcfurbstba wbtc j ogf-
gfbicmorcool eftj-cfurbstbas.
=.
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Qwjre Bgtfhhbifgaf bs jg bggovjtbvf lbstrbmutfl bgtfhhbifgt pjrjlbie `or 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` `ool 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 `ohhowbgi lfsarbptbog (Hjkkjkuhj, Qwjpgj, ]jeyj,
& Ejcftcj, =>:
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`oreuhjtfl `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 `ore
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 `ool 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 `ougl 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 mf`orf ^QO, tcf ejbg `oaus o` tcbs rfsfjrac, wbhh mf
lbsaussfl.
=.:>m). \cf blfj mfcbgl bt aoefs `roe tcf wjy jgts `bgl tcf scortfst wjy to j
`ool lfposbt. Jt tcf bgbtbjh stjif, tcfy jhh wjglfr jmout rjgloehy hookbgi `or `ool. Js tcfywjglfr jrougl, tcfy lrop pcfroeogf og tcf irougl. Ogaf jgy o` tcfe `bgls `ool, bt rfturgs
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:2
to tcf gfst. Bt tcfg rfturgs jijbg to tcjt sjef `ool souraf, mjsfl og tcf prbgabphf tcjt jgts
jhwjys prf`fr to `ohhow tcf safgt o` pcfroeogf, jgl tcf strogifr tcf safgt, tcf cbicfr tcf
acjgaf tcjt tcfy wbhh `ohhow bt. Cowfvfr, my tcf gjturf o` pcfroeogf, bt fvjporjtfs ovfr tbef.
\cfrf`orf, ovfr tbef, b` lb``frfgt jhtfrgjtf routfs cjvf mffg `ougl to tcf `ool 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 `or 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 `or 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) `or 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 `ugatbog 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 `ugatbog, jgl tcf sohutbogs cjvf soef
aogstrjbgts mbglbgi tcfe. \cfrf`orf, sohvbgi j AO promhfe bgvohvfs `bglbgi tcjt sohutbog
jeogi tcf sft o` `fjsbmhf sohutbogs wcbac cjs tcf ebgbeue omdfatbvf `ugatbog 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 pfr`ore
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 `ougl so `jr, ajhhfl tcf pfrsogjh mfst or pmfst
- tcf `btgfss vjhuf o` bts pfrsogjh mfst
Gow, `or 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 `or tcf ahjssbajh (orbibgjh) ^QO jrf js `ohhows4
+ 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 `or
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 `or irfjtfr aogtrohog sfjrac jmbhbty, rfafgt vfrsbogs o` ^QO usf lb``frfgt vjhufs, wbtc a: + a= 5
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==
a= 5 soabjh pjrjeftfr, aogtrohs cow euac tcf pjrtbahf eovfs towjrls tcf ihomjh mfst, js
fxphjbgfl jmovf.
r: & r= 5 rjgloe vjrbjmhfs ugb`orehy 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 `ohhows4
Jhiorbtce B4 ^jrtbahf Qwjre Optbebzjtbog
:4 ]jgloehy bgbtbjhbzf swjre
=4 wcbhf(tfrebgjtbog arbtfrbj got eft) {
14 `or(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
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=1
fgl o` tcf proafss, tcfrf wjs j gffl to aogtroh tcf vfhoabtbfs so tcjt tcfy mfaoef sejhh to
fgaourjif fxphobtjtbog jrougl rfibogs jhrfjly `ougl 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 `or 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 `ougl 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 Bg`orefl ^jrtbahf Qwjre (HB^Q) mjsfl
og sft-tcfory. \cfy ajhhfl tcf jhiorbtce Qft-mjsfl HB^Q (Q-HB^Q). \cfy tcfg aoepjrfl tcf
pfr`orejgaf o` Q-HB^Q og soef mfgacejrk \Q^ bgstjgafs wbtc tcjt o` Q-AH^QO, jgl `ougl
tcf rfsuhts to mf proebsbgi, fspfabjhhy bg tcf ajsf o` hjrif sajhf promhfes. J`jq & Qjbgb
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=:=) jhso prfsfgtfl j Lbsarftf ^jrtbahf Qwjre Optbebzjtbog (L^QO) tfacgbquf to sohvf
irjpc-mjsfl \Q^, tfstfl bts aogvfrifgaf, jgl `ougl tcf rfsuht `jvourjmhf. Yu ft jh (=>:1)
proposfl j lbsarftf ^QO (L^QO) jhiorbtce wbtc saout pjrtbahfs to sohvf tcf promhfe o`
sfhfatbgi ejtfrbjhs `or 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 pfr`orefl 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 `or \Q^, bt wjs lbsaovfrfl tcjt gogf o` sffefl to
tjkf jlvjgtjif o` tcf bg`orejtbog 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 suaafss`uhhy jpphbfl to j vfry wblf rjgif o`
`bfhls bgahulbgi aoembgjtorbjh optbebzjtbog promhfes, `ugatbog optbebzjtbog, `bglbgi optbejh
routfs, sacfluhbgi, struaturjh optbebzjtbog, bejif jgjhysbs, ljtj ebgbgi, ejacbgf hfjrgbgi,
mbobg`orejtbas, eflbajh bg`orejtbas, 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).
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=6
1
ACJ^\F] 14 FAL^QO JHIO]B\CE
1.: Bgtroluatbog
\cf lbsarftbzjtbog o` tcf ^jrtbahf Qwjep Optbebzjtbog (^QO) tfacgbquf `or tcf \rjvfhhbgi
Qjhfsejg ^romhfe (\Q^) bg tcbs ajsf bs suac tcjt j posbtbog bs j afrtjbg pfreutjtbog o` flifs
tcjt `ore j Cjebhtogbjg ayahf. Cfgaf, b` `or fxjephf, wf jrf aogsblfrbgi sbx abtbfs, jhh
bgtfraoggfatfl bg j efsc, tcfg j posbtbog aouhl mf tcf sft o` flifs4 {(:, =), (=, 1), (1,
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=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
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=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, `or 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 `ugatbog 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 `ugatbog 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 `or 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. \cfrf`orf, fqujtbog (=)
mfaoefs4
+ 5
+ (1)
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=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 `ore 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 `ore
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 ajrf`uh got to sfhfat jg flif tcjt wbhh
cbglfr j Cjebhtogbjg ayahf `roe `orebgi my jlcfrbgi to tcf `ohhowbgi ruhfs4
Bg tcf `bgjh hbst o` flifs to aogstbtutf tcf gfw posbtbog, fjac golf eust jppfjr fxjathy twbaf.
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=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 `orefl fxafpt bt bs tcf hjst flif, b.f.
prfvfgt tcf `orebgi o` sum-tours.
Lf`bgbtbog 34 @btgfss @ugatbog bs lf`bgfl js tcf sue o` tcf aosts o` jhh tcf flifs tcjt `ore
tcf gfw posbtbog.
Lf`bgbtbog 24 \cf vjhuf o` G bs tcf guemfr o` pjrtbahfs bg tcf fgtbrf sfjrac spjaf.
\cfrf`orf, tcf gfw upljtf fqujtbogs `or 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 `ore j posbtbog, bs
\cfrf`orf, wf aogsblfr bt rfjsogjmhf to sft tcf vjhuf
Z 5 (
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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 `ugatbog, wcbac
tjkfs bgto aoigbzjgaf jhh otcfr flifs tcjt ejkf up tcjt posbtbog. \cfrf`orf, 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 `ore j Cjebhtogbjg ayahf wcbac wbhh mf sfhfatfl `roe
jeogi tcf gflifs bg FlifMjgk.
\cf promjmbhbty o` jg flif mfbgi sfhfatfl `roe g flifs bs
. \cfrf`orf, tcf promjmbhbty o`
sfhfatbgi jg flif jeogi tcf e flifs to mf sfhfatfl `roe tcf gflifs bg FlifMjgk bs
.
\cfrf`orf, 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
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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
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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 >.
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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 >.>
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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,
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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,
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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
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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
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12
@roe tcf irjpc, bt ajg mf sffg tcjt tcf jvfrjif guemfr o` btfrjtbogs bgarfjsfs stfjlbhy wbtc
bgarfjsf bg guemfr o` abtbfs `or 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 pfr`ores 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 `or 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.
\cfrf`orf tcf rfsuhts mfaoef eorf jgl eorf ugrfhbjmhf js tcf guemfr o` abtbfs bgarfjsfs.
\cbs irjpc ibvfs j vfry ahfjr pbaturf o` tcf pfr`orejgaf o` tcf tcrff vjrbjgts. \cf FA^LO
vjrbjgt pfr`ores shbicthy mfttfr tcjg tcf FAL^QO(\)vjrbjgt, wcbhf tcf FAL^QO(M) vjrbjgt
cjs j vfry poor pfr`orejgaf aoepjrfl to tcfe. \cbs rf-fg`orafs 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
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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
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>% 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
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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
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6:
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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 >;moohfjg`ougl 5 `jhsf;wcbhf(lueeyHbst.cjsGfxt()) {
b`(!`ougl) {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();`ougl 5 truf;aougt 5 >;
} fhsfb`(lueey.ift@roe().fqujhs(gfxtFlif.ift\o())) {lueeyQtrbgi 5 gfxtFlif.ift@roe();lfstbgjtbog 5 lueey.ift\o();lueeyHbst.rfeovf();`ougl 5 truf;aougt 5 >;
} fhsfb`(lueey.ift\o().fqujhs(gfxtFlif.ift@roe())) {lueeyQtrbgi 5 gfxtFlif.ift\o();lfstbgjtbog 5 lueey.ift@roe();lueeyHbst.rfeovf();
`ougl 5 truf;aougt 5 >;
} fhsfb`(lueey.ift\o().fqujhs(gfxtFlif.ift\o())) {lueeyQtrbgi 5 gfxtFlif.ift@roe();lfstbgjtbog 5 lueey.ift@roe();lueeyHbst.rfeovf();`ougl 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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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 >;
-
7/25/2019 BSc Thesis - An Enhanced Communication Discrete Particle Swarm Optimization Algorithm for the Travelling Sales
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}
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
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