Simulating a Mathematical. Representation of Stochastic Combat. Model

Transcription

1 Smulag a Mahmacal Rprsa f Schasc Cmba Mdl Asss rf Dr Saad Talb Hass Uvrs f Babl - acul f Scc - cmpurs sccs dparm Iraq Absrac Th ma purps f hs papr s buld a M Carl smula mdl whch ca b usd smag h ms mpra facrs a suggsd schasc cmba ar mdl I ams h fasbl rprsa f h mahmacal aspcs f h hr f hmgus schasc cmba mdls Ths vlvs h sa spac rprsa h ac Klmgrv quas f h hmgus cmba mdl h cdal kll ras r-kllg m dsrbu h umbr f survvrs ad h rma dcs ruls usd ad h sa prbabls avalabl Kwrds : Markv Ar mdls Klmgrv quas Smula Irduc M Carl smula s a ffcv Opraal Rsarch l ha hlps h dcs makrs pcg hr suabl ppsg dcss I als prvds a abl dscuss h pras s prs prdc h fuur r aalz wha has ccurrd h pas Smula ffrs ral-m aalss f h mplcas f frdl ad m dcss [] I s vr mpra ffr h dd frma abu ach assumd cmba rdr asss h accal dcss f h cmmadrs Curss f ac a cmba ca b smulad a rcurrc mar prvd frma h cmmadrs abu h

2 pcd curr ad fuur bhavr f a parcpa a m-ffc mar Ths frma hlps h cmmadr mak h rgh dcss a h rgh m Th schasc smula mus b ru ma ms g a da f h varabl f casuals ad hr rsuls Th m sp a sa s radm varabl whs ma s h rcprcal f ha ra A smpl v-sp M Carl smula s hus pssbl b usd T mpl h schasc mdl mus gra h m hsr a m-sp M Carl smula ha crms m ul sm rma cd s rachd A smula f ar mdls s a cmpur rprsa f h dvlpd sua sas f w ppsg frcs usg daa ad prcdurs dsgd dpc a acual r assumd ral-lf sua Ar Mdls Ms ar mdls dfd b mas f lsss prsl waps ad qupm as a rsul f cmba pras Th ssal prblm facd applg h ar mdls s hw assg h varus cffcs h mdl quas ad hw mach hs cffcs rach a apprmal crrc aswr abu h fuur bal [ ] Ar ra s a facr ha dpdg h frg ra f ach sd durg ach m ad masurd b h cur ffcs I mas ha sd s makg a kll durg h m rval gv ha h cmba s sa a m [3 4] Th rm r S mas: h prbabl ha h cmba s sa a m r S Th kll ra f sd a m s dfd [4 ] as: W r{ K lm 0 Whr k ; mas ha maks a kll durg ; If s small ugh h: r{ K } W

3 r{ K V} W V lm 3 0 r{ K [ K } 4 r{ K S } 5 whr: d : b h kll ra f h sd d a m gv ha maks klls up m ad: d Th prss fr was drvd b J [4 ] as : f 6 0 ad: f f f u du fr 7 u d u fr 8 0 Whr f s h prbabl ds fuc f h r-kllg m T dvlp a cra prss fr h kll ra q 6 usg h apprpra dsrbu fr such prcss I s vr mpra prps a mhd drv a prss fr h kll ra frm cra dsrbu us ur smula mdls Th pal dsrbu s h suggsd dsrbu fr such prcss Th suggsd mhd s as fllws: Th r- kllg m has a pal dsrbu wh a ma f ad havg h fllwg ds fuc: f 9 Usg rpral Dcs Rul quas [5] w ca fd: f u 0 u du ; 3 3 ; f 3

4 4 f 3* ; f! ad f! 0 u d u 0 du u u 0 3 3! B subracg a w succssv fucs w ca fd: ;! 3 ; 3! Ad s ;! Subsu hs valus q6: f K Th fal prss fr h kll ra was fud b: k!! Saad [5] shws ha h kll ra f ach sd s h sam as h hazard ra h Rlabl hr ad qual ma Ths dvlpd prss ca b usd h smula apprach gra a kll ra a ach sag f h cmba

5 3 Klmgrv Equas T cmpu [ 6] b a f: b a K a k b K a k a k k k ba k a bk b k Th Klmgrv Dffral quas: m m ad: j k kj 3 k As dscr m[ 6]: S s j k k kj Ad mar rms; S s Sg ds gvs; s ds s ds ; r s ds s s[ ds I] ; Whch Gvs: s s Q ; Whr: Q ad I Ths s calld h klmgrv frward qua S ds j S ds j s k s k k d ds slu s k j S k q ds q s j If : q q h s k kj k j s k k ug S d : h Chapma-Klmgrv quas: kj q kj Q ; S s hrcall pssbl slv Klmgrv quas a p Q Q! 4 kk j kj 5

6 6 Usg h abv dfs h ac klmgrv quas fr h cmba sas ca b drvd [4 ] as fllws: a r h al sa : ] ][ [ ] [ ] [ ] [ ] [ lm 0 ] [ d d 5 b r h scd sa whr f < < whch s h ars h al sa sd klls frm Y sd frs: ] ][ [ ] [ ] [ ] [ lm 0 ] [ ] [ d p d a 6 c r sas whr f < < Th hr sd ars sa r Y sd klls frm h sd frs ] [ ] [ d p d 7 d r sas whr f < < ad f < < hs s calld Tras sa ] [ ] [ ] [ d p d 8 5- r sas f whr f < <

7 d p d f [ f 9 6- r sas f whr f < < 0 d p f [ f ] d 0 f Th al cds ar 0 0 fr all hr sas 4 Th prblm O frc calld hrag ahr frc calld Y Th ar plag fac ach hr a ar cmba Th mus prpar hr frcs fr acual war Ma accal facrs mus b crrcl aalzd bfr h bgg f h cmba r durg s al sags achv h vcr r a las mmz h lsss ad prvs h m frm achvg hs bjcvs Thr ar ma pssbl mlar alravs fr ach sd Th aackr has a surprs aack hs pp b chsg h m ad h alravs f aack Each sd maks a dcs abu h allca ad h dffus f hs frcs h parg dscpl h ffcv srs ad h suabl p f arcraf's ha mus b usd frsl mamz h pcd casuals h m's frcs Th ms mpra accal dcs h mus rach s hr aack r dfd l Ths dpds h avalabl rlabl frma abu h cduc f h ar cmba phass Th cmmadrs ach sd cduc plag h ar cmba Th alwas sarchg fr h ws ad ffcv was dsr r dfa h m sd rduc hs bfs h mmum pssbl lvl Th als d vrf ad prv h vald ad h crrcss f hr plas 7

8 5 Th smula mdl A M Carl smula mdl was bul smula hs prblm llwg s h mpra frma rqurd: Th al frc f ach sd whch bulds ad frmas h cmba pwr Th ar r kll-ra f ach frcs h ffc f ach frc h hr Ths s smad dpdg h frc pwr f ach sd 3 Th accs usd whch cas h rma dcs ruls 4 Th ffcs f h al casuals 5 Th pcd dura f h cmba ad s dva 6 Th pcd umbr f h survvd arcraf's fr ach sd 7 Th pcd wg prbabl ach sua Th cmba sa ca b dfd a a m as whr rprss h rmag arcraf's -sd ad h rmag umbr h Y-sd Th al sa s ad h fal sa s hr f r f whr grar ha f 6 Mdl assumps Th basc assumps f h suggsd mdl ar as fllws: Tw ar frcs ad Y ar gagd a cuus ar cmba All arcraf h sam sd ar apprmal dcal; ha s h hav h sam rkllg m dsrbu 3Th m ach arcraf aks damag s args calld rkllg m s a arbrar -gav cuus radm varabl 4Th grud srvcs ful ad ammus suppls ar ulmd 5Each arcraf rcvd fr r mr durg s mss calld dsrd ad rducd frm h al avalabl arcraf's umbr 8

9 6Wh h umbr f rad arcrafs h sd rachs s brak p f r h umbr f rad arcrafs h Y sd rachs s brak p f h bal wll rma 7Th r-kllg m f ach sd s smulad afr ach sa f h bal Ths m was grad usg apprpra radm varabls wh ma qual h al kll ra mulpld b h frc sz f h ppsg frc a ha m 8 Th w ar frcs ad Y ar gagd wh al frc h sd ad h Y sd Afr m rval h w frcs wll b f h fllwg sas: - ad - dpdg h al kll ra f ach sd ad wh wll mak a kll frs Th kllg vs h ms prvus cmbas ar sd [4 ] Th r-kllg m was pall dsrbud as prvd ms lraur [ 4 7 ] Th kll ra cdg h sa f h w frcs ad h m f h bal was smulad durg h ru f h smula prgram as h al kll ra mulpld b h frc sz a ha m h fllwg qua was usd smula such prcss Udr amd fr cds h ar ra f dpds hw ma 's ar shg a hm ad lkws fr [8] d d β d ψ β s d a ar ra cffcs prssd rms f casualsy frr m Ad Ψ s a ar ra cffcs prssd rms f Y casuals frr m 9 Th prgram rqurs h al kll ras f h w frcs βψ h al frcs 0 0 ad h brak p f f as pus 0 Th pcd rval f m ha ach frc wll rmas a sa a a m was drmd Th pcd dura f h bal T D s h summa f 9

10 hs ms fr ach sd Ad h prbabl f bg hs sa s calculad b dvdg h m f bg h sa vr h al dura T D Wh h bal prcds h w frcs wll b rducd a mar dpdg h cdal kll ra whch dpds h frc ad m hs kll ra wll b zr wh h frc rachs s brak p f r f bcaus h bal mus b rmad a hs p I hs mdl a apprma rla s dvlpd sma h cdal kll ra f ach frc ach sag f h cmba dpdg h curr sz f hs frc Ths rla s as fllws [5]: Kll ra a ha m al kll ra * sz f h ppsg frc a ha m β ψ 3 Ths rla s a smula quval qua Th r-kllg m was grad usg h qua blw [ 90 ]: T - kll ra *LOG RND R 4 Th basc sps ha usd smula hs prblm ar as fllws: - Gra w radm umbrs ad h us hm gra w radm varabls usg qua 3 rprs h r-kllg m f h -frc ad h -frc frm h pal dsrbu - Cmpar hs w ms ad slc h smalls hs mas ha hs frc wll lav hs sa fasr ha h hr hr wrds "hs mas ha hs frc wll lss r rduc b bfr h hr" r h frs kll wll ccur hs frc - Th sa wll b grad b rduc h dcad frc h lsr b ad sa h hr frc as s - Rpa hs prcss ll f h w frcs rachs s brak p valu f r f 0

11 - Th m dd fr ach frc rasm frm sa sa was dcad ad summarzd rprss h al cmba dura T D Ths m s quval h fllwg qua Lachsr mdl [] Whr h m a whch s f s hrfr lg ; g p βψ ψβ - Th prbabl f bg ach sa was als calculad b dvdg h m ach sa b h al m Ths valus rprss h smula rsuls f h s f Klmgrv quas - Th rs umbr f ach sd afr frc rma dcas h umbr f survvrs whch s s ad s as dcad h fllwg qua [8] s ψ 0 0 β f 7 Mdl rsuls Th smula rsuls ffr mr frma ad dals h dcs-makr ha h Lachsr rsuls Lk h frc sz a a m durg h cmba h pssbl sas f h cmba h wa ha cmbas bg cducd ad h m ad h prbabl f rasmg frm sa sa llwg ar h smula rsuls fr w accal scars as shw Tabls ad Dffr suggsd al frcs ad kll ras ar usd ach scar Each abl csss f w pars: I par a h sag m ach sa bfr rasg h sa s shw whl par b shws h prbabl f bg ach sa durg h cmba dura Th umbrs ach arra fllwd frm h p-rgh rhas crr h bmlf Ths umbrs rprss h m ha ach frc sag ha sa h arra f par a ad h prbabl f sag ha sa par b arra Th frs umbr h p r rhas crr f par a arra rprss h m ha h cmba sa hs al

12 sa ad rprss h prbabl f bg hs sa par b arra Th umbrs ach arra dcas h pssbl sas f h cmba frm s al sa h Brak Sa whl h zrs dca h -vsd sas Ths umbrs als fllws h squc f h pssbl vsd sas ach cmba scar hs wa dcas h ras sas f hs cmba rm h fal p r h lws ps umbr h survvd umbr f ach frc ca b fud drcl add h pcd cmba dura Ths arra dcas h advaags f smula wh rspc Lachsr umrcal mdl rsuls Tabl Tms ad prbabls f bg ach sa fr h suggsd Scars: Scar I -frc5 -frc0 f 5 f 3 -kllg ra04 -kllg ra04 Y *** cmba dura a: STAYING TIME IN EACH STATE DURING THE COMBAT DURATION

13 **** Tal prbabl b: ROBABILITY O BEING IN EACH STATE DURING THE COMBAT Scar II 0 4 ; Y 0 Y kllg ra05 Y kllg ra03 **** Cmba dura 3350 A: STAYING TIME IN EACH STATE DURING THE COMBAST DURATION 3

14 Y **** Tal prbabl B: ROBABILITY O BEING IN EACH STATES DURING THE COMBAT DURATION Th fllwg vr mpra cmba paramrs ca b valuad Th smula prgram was ru 000 ms rdr rach a sad sa Ifrma was cllcd fr h ru whch gav h fllwg rsuls - Th ma ad sadard dva f h bal dura m T D - Th mas ad sadard dva f h umbr f survvrs f ach sd 3- Th wg prbabls fr ach sd [] ad [Y] Sm f hs cmba paramrs ca b valuad usg h aalc mhd Lachsr mdl cmpar wh h smula rsuls ha bad frm h 4

15 smula mdl dvlpd hs papr fr hs prblm Th smula rsuls ar valuad fr hr smula mdls as a smula Lachsr mdl drcl wh u chagg h kll ra wh m ad frc sz durg a cmba dura a ru mdfd smula mdl whch h kll ra was vard dpdg h frc sz ad 00 rus mdfd smula mdl Th las mdl was r-rus 00 ms rdr fd h dd sascal paramrs ha dcad h las clum f ach abl Ths rsuls ar dcad abl's ad 3 fr dffr suggsd accal suas 5

16 Tabl ; Avrag cmba paramrs fr h accal sua II f 5 f 5 -kllg Ra -kllg Ra Aalc Smula Mdfd Mdfd rsuls O-ru Smula Smula paramrs lachsr lachsr O-ru 00-ru AVERAGE COMBAT DURATION STDEVO COMBAT 38 DURATION AVERAGE NUMBER O SURVIVAL Y ST DEV NUMBER O 396 SURVIVAL Y AVERAGE NUMBER O SURVIVAL STDEV NUMBER O 347 SURVIVAL ROBABILITY THAT Y WILL 06 WIN ROBABILITY THAT WILL 94 WIN 6

17 Tabl 3 Avrag cmba paramrs fr w accal sua f 5 f 5 -kllg Ra 5 -kllg Ra Aalc Smula Mdfd Mdfd rsuls O-ru Smula Smula paramrs lachsr lachsr O-ru 00-ru AVERAGE COMBAT DURATION STDEVO COMBAT DURATION 499 AVERAGE NUMBER O SURVIVAL Y 54 4 STDEV NUMBER O SURVIVAL Y 43 AVERAGE NUMBER O SURVIVAL STDEV NUMBER O SURVIVAL 8 ROBABILITY THAT Y WILL WIN 8 ROBABILITY THAT WILL WIN 9 Tabl 4 blw summarzs h rsuls f dffr pcd accal suas f h w ppsg frcs ad hr kllg ras I par a f hs abl h aalc rsuls f ach accal sua was calculad usg Lachsr mdl Th smula mdl gvs rlabl rsuls a asr mar ad ffrs h ms dd frma h dcs-makr as shw par b f h sam abl A cmpars bw hs rsuls s usd hlp h aals chsg h pmal mhd slv such prblm as w kw h chqu ha usd slvg h prblm s vr mpra praal rsarch apprachs Th dvlpd smula mdl ca assss h dcs-makr plag hs accal dcs 7

18 fuur f h uss acual daa Th dcs-makr ca appl hs mdl drcl dpds h avalabl m r h rmag m h cmba bgg Tabl 4; Summar f h ar cmba paramrs sascs fr h ms prbabl suas Sd Sd Y-KILL -KILL COMBAT Y Y RATE RATE Dura Survvrs Survvrs ORCE ORCE r da r da T D DAYS s s A: Lachsr aalc rsuls fr dffr pcd accal suas Sd Y Sd Y Kll Avrag Sadard Avrag Sadard Avrag Sadard W rc ra Kll Dura Dva umbr f Y Dva umbr Dva rbabl Y 0 rc Y Ra Tm Of Survvd Of Y f Of 0 ET D dura EY survvd survvd Survvd Y δ T δ Y E δ w W B: Smula rsuls fr dffr pcd accal suas 8

19 8 Cclus Th mdl ucms rprs h fuzz r urbul facrs ha a dcs-makr was g sm ral r apprma dcas abu hm asss hm s hs accal dcss durg r bfr h cmba bgg Ths maks hm pmz hs hugh cras h ppru f achvg h cmba bjcvs Durg h sud f mahmacal rprsa f cmba mdls was fud ha hs mdls ar ffc dvlp a dcs ad h mlar cmmadrs Hc was vabl dvlp a smula mdl ha assss a mlar dcs-makr makg hs ffcv dcss h suggsd smula mdl wll ffrs ; h shap h m ad h prbabl fr ach vsd sa f h w cflc frcs a lac frm h pcd avrag ad sadard dva f h cmba dura h pcd ad sadard dva f h umbr f survvg us f ach sd a h d f h bal h pcd wg prbabls f ach sd h ras prbabls r h shap f h cmba sa rass wh m ad Th ar curv r h rla bw h avalabl umbrs f h w frcs a a cmba m Rfrcs []: Abraham M ad Hlma I S 985"Opmal Rfulg Srags fr A Md - Vhcl l" Naval Rs Lgscs Quarrl vl 3 pp []: Lwll B A99"Ar rmulas fr Drmsc Mdls f Larg-Scal Cmba" ublshd : Brack J Krss M ad Rshal R warfar mdlg Jh Wl & Ss MORS Ud Sas [3] Bragr DA"A Quaav Cmpars f Varus Scdar rcssg Mhds fr Th urps f Trackg Lw Targs " hd hs Uvrs f Da Ud Sas 995 9

20 [4] J Yag " A as Apprma f Hmgus Schasc Cmba" hd hss Uvrs f suh Carla Ud Sas 99 [5]: Aljbr Saad T Hass ; "A Crbu Smulag a Taccal Dcs fr ar Cmba Mdlg" hd hss mlar cllg f grg Iraq Baghdad 00 [6]: Bha 97" Elms f Appld Schasc rcsss" Jh Wl & Ss Ic [7]: Grdv V 997 "udamals rcpls f Ar Dfs Bals Mdlg :A Lcur " Mahmacs Smula : Tchlg cr " Russa " Tl [8]: Davd L Brs 99"Effc Ccra f rcs r Hw gh Ouumbrd ad W" ublshd : Brack J Krss M ad Rshal R warfar mdlg Jh Wl & Ss Ic MORS Ud Sas [9]: shma G J" Ccp ad Mhds Dscr Ev dgal Smula " Wl Nw Yrk 973 [0]: Harrs M &Grss" udamals f quug hr" Scd d Jh Wl & Ss Ic USA 974 []: Jaswal NK ad Nagabhushaa B S99 "Trma Dcs Ruls Cmba Ar Mdls " ublshd : Brack J Krss M ad Rshal R warfar mdlg Jh Wl & Ss Ic MORS Ud Sas