22nd Inrnaional Congrss on Modlling and Simulaion Hobar Tasmania Ausralia 3 o 8 Dcmbr 2017 mssanz.org.au/modsim2017 Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc J.R. Lookr a V. Mak-Hau b and D.O. Marlow a a Join and Opraions Analysis Division Dfnc Scinc and Tchnology Vicoria b School of Informaion Tchnology Dakin Univrsiy Vicoria Email: jason.lookr@ds.dfnc.gov.au Absrac: This papr prsns a fl managmn modl for miliary aircraf ha gnras daily flying and mainnanc allocaions. This modl has wo novl faurs ha disinguish i from prvious modls of fl managmn in h liraur. Mos noably random unschduld mainnanc is ingrad wih an opimisaion approach. This is imporan as unschduld mainnanc is a rgular occurrnc in day-o-day miliary aircraf fl managmn hus limiing h abiliy of drminisic modls o provid usful guidanc. Furhrmor h modl is dsignd o discovr policis ha dynamically adap o random vns o dlivr opimal fl srvicabiliy. Ohr faurs of h modl includ flying-hour basd phasd mainnanc wih variabl inducion (day whn mainnanc bgins) day-basd rgular inspcions and spara phasd and fligh-lin mainnanc faciliis wih boh manpowr and lin capaciy consrains. In h modl srvicabl aircraf fly up o a spcifid nbr of hours pr day. Onc a crain nbr of flying hours has bn achivd hos aircraf mus bgin phasd mainnanc wihin an allowabl inducion rang. Rgular inspcions ar an addiional yp of schduld mainnanc basd on lapsd im. Phasd mainnanc ypically aks a fw hundrd mainnanc man hours whras rgular inspcions ak ns of man hours. Two yps of mainnanc faciliis ar includd. Phasd mainnanc can only b undrakn on h phasd mainnanc lin and rgular inspcions ar ypically undrakn on h fligh lin. If a rgular inspcion is du a h sam im as a phasd srvic hs can b prformd concurrnly. Unschduld mainnanc is undrakn in fligh lin mainnanc if incurrd on srvicabl aircraf or in ihr faciliy if discovrd during h cours of schduld mainnanc. W us Approxima Dynamic Programming (Powll 2011) o formula h dcision making modl. Th objciv is o find h policy ha maximizs h xpcd valu of h conribuion funcion. Our conribuion funcion sks o maximiz h flying hours achivd and mainnanc hroughpu of h fl ovr a im priod wih a wighing facor balancing hs prioriis. Dcision variabls includ h allocaion of flying hours and mainnanc man hours pr aircraf pr day and whn aircraf ar inducd ino phasd mainnanc. Th modl consiss of 75 ss of consrains dscribing: h allocaion of flying hours o srvicabl aircraf and mainnanc man hours o aircraf in mainnanc; h allocaion of diffrn mainnanc yps o aircraf in diffrn mainnanc faciliis; and sa ransiion ss o drmin whn aircraf ar ligibl or rquird o bgin or nd yps of mainnanc. Th unschduld mainnanc consrains includ an xognous sochasic variabl ha rprsns h daily amoun of nw unschduld mainnanc gnrad randomly from a lognormal probabiliy disribuion. Aircraf ar allowd o carry forward a limid amoun of unschduld mainnanc whil rmaining srvicabl. Howvr onc a spcifid lvl of unschduld mainnanc is xcdd h aircraf mus undrgo mainnanc a an appropria mainnanc faciliy unil h ousanding unschduld mainnanc is rducd o a rquird lvl. In Approxima Dynamic Programming a policy is a funcion ha rurns a dcision givn h currn sa of h fl (in h prsn conx). W considr a drminisic look-ahad policy ha dpnds on a discoun facor and a look-ahad im horizon. W undrak compuaional xprimns wih a squadron of 12 aircraf ovr a 180 day priod wih im horizons ranging from 1 o 14 days and discoun facors from 0.8 o 1. Indicaiv simulaion rsuls ar prsnd whr h im horizon subsanially dominas h discoun facor wih rspc o influnc on h man oal flying hours achivd. Whil longr im horizons produc largr opimal objciv valus for a givn paramr s and random daa simulaion rsuls may no significanly diffr whn compard wih shorr im horizons du o random variaion. This is apparn in our rsuls dmonsraing h ponial for subsanial savings in compuaional im. Kywords: Aircraf fl managmn compuaional sochasic opimizaion unschduld mainnanc 1392
Lookr al. Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc 1. INTRODUCTION 1.1. Miliary aircraf fl managmn Managing a fl of miliary aircraf is an inhrnly complx ask. Th lif of a miliary aircraf fl is ypically 20-30 yars. Ovr h fl lif hr ar nrous rquirmns o b m boh by h whol fl and for h individual squadrons wihin h fl. Evry yar ach squadron will nd o m a nbr of flying rquirmns for insanc o rain aircrw and m opraional rquirmns. Furhrmor if hr ar mrgncis such as naural disasrs govrnmns may call on h miliary o assis a which im i is xpcd ha h miliary will b abl o rspond as rqusd. Consqunly hr is an implici rquirmn for squadrons o hav a sufficin nbr of aircraf availabl o b dployd a shor noic in such siuaions. Ths flying rquirmns nd o b achivd whil also ming h fl s mainnanc rquirmns. Th schduld mainnanc rgim consiss of svral yps basd on boh lapsd im and achivd flying hours. Rgular inspcions ar boh im-basd (.g. daily wkly c.) and flying hours-basd and usually ak 1-2 days o compl; only im-basd rgular inspcions ar considrd hr. Phasd mainnanc is flying hours-basd and may ak a fw hundrd mainnanc man hours (MMH) o compl quaing o a priod of days or wks. Finally dp or dpo mainnanc usually occurs vry fw yars or svral hundrd flying hours and may ak wks or monhs o compl; his is no considrd in h prsn modl. In addiion unschduld mainnanc ariss rgularly o disrup h plans mad by fl managrs. Unschduld mainnanc rquirmns may b discovrd bfor an aircraf is schduld o fly hus rndring h aircraf unsrvicabl or i may b discovrd during h cours of schduld mainnanc hus dlaying h rurn of h aircraf o a srvicabl sa. Diffrn mainnanc faciliis handl h diffrn mainnanc yps and som yps of srvics may b combind. Givn all of hs facors fl plannrs and managrs mak daily dcisions abou which aircraf o fly and how much and which o mainain and how much in ordr o m ongoing rquirmns. 1.2. Liraur rviw Th liraur in his ara is rlaivly nw compard wih civilian applicaions. Civilian and miliary aircraf fl managrs hav diffrn objcivs and consrains. Civilian aircraf fl managrs ar ypically concrnd wih maximizing profis wih aircraf flying poin-o-poin bwn various ciis. Th objcivs in miliary aircraf fl managmn ar mor concrnd wih maximizing aircraf availabiliy or h nbr of flying hours achivd wih dcisions rgarding which aircraf o fly ach day and how much. Mos of h liraur on miliary aircraf fl managmn concnras on purly drminisic applicaions using mahmaical programming chniqus o dvlop a flying and mainnanc plan ovr a givn im horizon. A Masrs hsis from h Naval Posgradua School (Pippin 1998) dvlopd a mixd ingrlinar program (MIP) in ordr o minimiz h dviaion from h goal lin for a US Army hlicopr fl. Th goal lin is a visual planning aid ha racks and sors individual aircraf and hir flying hours rmaining unil hir nx phasd srvic o nabl a sady flow of aircraf ino mainnanc. This has bn xndd (.g. Kozanidis al. 2012) by sking o gnra flying and mainnanc plans ovr mulipl im horizons (hr a monhly plan for six monhs) whil maximizing aircraf availabiliy (i.. h oal fl hours rmaining unil phasd mainnanc). Ths auhors (Gavranis and Kozanidis 2015) hn xploid h limid nbr of aircraf combinaions ha can bgin and nd phasd mainnanc o dvlop an xac soluion o his problm. A US Coas Guard sudy (Hahn and Nwman 2008) usd a MIP o gnra an opimal wkly plan for 12 wks for a hlicopr fl. Phasd mainnanc is includd and hlicoprs may b dployd o wo spara locaions. Th objciv is o minimiz hlicopr movmns (.g. bwn dployd and non-dployd) wih pnalis for no ming flying rquirmns. Ohr paprs apply MIPs o plans ha considr daily rquirmns: on papr (Cho 2011) dvlops a wic-daily plan wih wo fixdsori yps for 15 aircraf ovr a on-yar horizon whil minimizing h maxim nbr of aircraf in mainnanc; whil anohr papr (Marlow and Dll 2017) dvlops a daily plan for various im priods (20 and 30 days) and aircraf nbrs (12 and 24) incorporaing boh phasd mainnanc and rgular inspcions wih h aim o balanc mulipl rquirmns including daily and oal flying hours rquirmns as wll as a yp of goal lin mainnanc saggr a spcifid days. Ohr work in his fild has includd unschduld mainnanc. Th iniial focus (Maila al. 2008) has bn dvloping discr-vn simulaion modls incorporaing various yps of schduld mainnanc wih h focus on avrag availabiliy. Ths auhors nx apply a rinforcmn larning approach (Maila and Virann 2011) in ordr o ihr maximiz h availabiliy ovr an infini im horizon or xcd a 1393
Lookr al. Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc arg availabiliy lvl ovr a fini im horizon in conflic condiions. Thy s various opimizaion policis basd on drmining h bs im o induc an aircraf ino mainnanc (which may includ dlaying nry). Thy also sk o dvlop a dcision suppor ool (Maila and Virann 2014) combining hir discr-vn simulaion modl wih a muli-objciv simulad annaling algorihm wih h aim o boh maximiz avrag aircraf availabiliy and minimiz dviaion from h dsird mainnanc schdul. 1.3. Aim of his work Th liraur rviw dscribs drminisic MIP modls usd o gnra flying and mainnanc plans. Whn unschduld mainnanc ariss hs plans will hav o b adjusd and h shorr h planning horizon h grar h impac on hs plans. Th simulaion modls survyd incorpora unschduld mainnanc bu do no focus on gnraing flying and mainnanc plans. Hnc h primary aim of his work is o prsn a modl ha sks o opimally manag a miliary aircraf fl whil xplicily including unschduld mainnanc. W hrfor rquir an approach ha combins simulaion (o modl random unschduld mainnanc) and mahmaical programming in ordr o drmin opimal policis for an aircraf fl similar o h sing of mainnanc policis dscribd prviously (Maila and Virann 2011). Such a modl can provid fl plannrs wih h abiliy o mak dcisions ha improv oucoms whil xplicily incorporaing uncrainy. All faurs of h prsn modl canno b dscribd in his papr du o spac limiaions; insad h ky lmns ar inroducd. 2. MODEL DESCRIPTION W considr a singl squadron conaining a fixd nbr of aircraf opraing in pacim. Aircraf may fly up o a maxim nbr of hours pr day whn hy ar srvicabl. Unsrvicabl aircraf (i.. hos in schduld or unschduld mainnanc) ar unabl o fly. W includ wo yps of schduld mainnanc: rgular inspcions (RI) (basd on lapsd im) and phasd mainnanc (PM) (basd on flying hours). Ths mainnanc vns may b combind whr possibl. Aircraf bcom unsrvicabl whn a RI is du whras aircraf may b inducd ino PM wih som flxibiliy around h spcifid flying hours sinc h prvious PM. Dpo mainnanc and srvics ha can b compld wihin a day and no affc h flying program ar no includd in h modl. Th diffrn yps of mainnanc ar prformd a spara mainnanc faciliis. PM occurs a h phasd lin mainnanc (PLM) faciliy whil rgular inspcions occur a h fligh lin mainnanc (FLM) faciliy. Whn PM occurs a h PLM faciliy RI (if du) may also b combind wih h phasd srvic. Schduld mainnanc mus b compld in blocks; i.. onc work on schduld mainnanc bgins i canno sop unil h mainnanc is compl. Aircraf quu for mainnanc if hr is no spar capaciy a a mainnanc faciliy. Unschduld mainnanc (UM) has wo componns: im bwn failurs and im o rpair (duraion). Boh ar modlld as lognormal probabiliy disribuions basd on prvious analysis of aircraf daa (Marlow and Novak 2013). UM may b undrakn a h FLM faciliy (ihr whn an aircraf fails or is combind wih a rgular inspcion) or a h PLM faciliy (ihr whn an aircraf ha is ligibl for PM fails or whn UM is discovrd on ha aircraf during a phasd srvic). Nw UM MMH rquirmns for ach aircraf ar rvald a h sar of ach day according o h inpu probabiliy disribuions. Aircraf ar prmid o accula a spcifid amoun of carrid forward unsrvicabiliy (CFU) in MMH whil rmaining srvicabl. Howvr onc h CFU MMH hrshold is xcdd hs acculad MMH bcom h UM rquirmn and h aircraf mus undrgo mainnanc in ordr o rurn o a srvicabl sa. All UM rquirmns ar considrd qually imporan in h prsn modl and hnc urgn UM rquirmns canno b idnifid. 3. MATHEMATICAL METHOD AND MODEL FORMULATION 3.1. Approxima dynamic programming W us Approxima Dynamic Programming (ADP) (Powll 2011) o formula h dcision making modl. ADP is a family of mhods for solving squnial sochasic dcision making problms by finding h bs policy (dcision funcion) from a givn class of policis wih rspc o an objciv funcion. Th prsn ADP formulaion incorporas aspcs of mahmaical programming including an objciv funcion 1394
Lookr al. Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc consrains and dcision variabls. In addiion i conains sa variabls (ha hold h informaion ndd o mak a dcision) a ransiion funcion (ha govrns how in his cas h sysm of consrains ransiion h sa variabls from hir currn valus o hir nx a ach im sp) and xognous informaion (in his cas h unschduld mainnanc). Th objciv funcion is max T C( S X ( S )) 0 whr C( S X ( S )) is h conribuion funcion and X ( S) is h policy (dcision funcion) indxd by in h admissibl s of policis for sa S. Thr ar many diffrn policy yps (choics of admissibl policis) including policy funcion approximaions valu funcion approximaions myopic policis and drminisic and sochasic look-ahad policis (Powll 2014). 3.2. Formulaion Th modl is dscribd by (for ss of aircraf a Aand im T in days): Dcision variabls which ak wo possibl forms. Firsly x is usd for coninuous variabls whr h suprscrip can b: f whn rprsning aircraf abl o fly; plmri and plm for aircraf in phasd mainnanc (undrgoing PM RI or UM rspcivly); and flmri and flm for aircraf in fligh n lin mainnanc (undrgoing ihr RI or UM). Scondly y rprsns binary variabls.g. ya is 1 whn an aircraf nrs phasd mainnanc and 0 ohrwis. Th car symbol (^) indicas h dcision variabl aks affc a h nd of h day; ohrwis i aks affc a h bginning of h day. Sa variabls which ncod h informaion rquird a a paricular im o modl a sysm from ha f im onwards. Ths also ak wo possibl forms: R for coninuous variabls.g. R for h flying hours accrud sinc h las PM; and S for binary variabls.g. S is 1 if an aircraf rquirs PM and 0 ohrwis. A car symbol indicas h sa variabl is masurd (or obsrvd) a h nd of h day ohrwis i is masurd a h bginning of h day. Paramrs whr c i rprsns coninuous valus wih unis and n i discr dimnsionlss valus. Du o spac limiaions w ar unabl o show h full formulaion in his papr. Howvr w will ndavour o dscrib h ky lmns. Th conribuion funcion is C( S x ) 1 w x 1 x x x 1 x x B B f f plm ri plm flm ri flm bl plm bl a a a a a a F plm flm a a T aa c cmax cmax 2 f Th conribuion funcion is a wighd s of flying hours (firs rm in parnhss wih wighing w ) MMH (scond and hird rms) and a pnaly for no prforming PLM and UM on an aircraf on conscuiv days. All rms ar scald such ha hy ar dimnsionlss.g. c F is h maxim flying hours pr aircraf pr day. Hnc h objciv funcion maximizs boh h flying hours and MMH achivd. Th MMH rms rmov a symmry ha ariss whn aircraf ar unsrvicabl byond h soluion im horizon. Th modl conains 13 A T dcision variabls 28 A T sa variabls and O(75 A T ) consrains. Ths consrains may b dividd ino hr main cagoris: Flying hour and MMH allocaions ha limi h allocad flying hours or MMH o wihin spcifid rangs. For phasd mainnanc hs ar plm plm c y min a xa c y whr max a y is a binary dcision variabl ha aks h valu 1 whn phasd mainnanc is bing undrakn. Exra consrains modl h combinaions of mainnanc yps a h allowd mainnanc faciliis. Mainnanc faciliy allocaions ha drmin whn and whr mainnanc can b undrakn. Th plm consrains ar gnrally of h form ya S ha sa an aircraf undrgoing PM mus b on a phasd a mainnanc lin (PLM). Du o h allowabl combinaions of mainnanc a diffrn faciliis mor complicad consrains ar usd:.g. flm ri Sa Sa Sa sas ha an aircraf mus rquir a las on of 1395
Lookr al. Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc an RI or UM o b in FLM and S flm S plm s S 1 forbids an aircraf from bing in mulipl faciliis a a a (and srvicabl) simulanously. Addiional consrains rack mainnanc capaciy o nsur his is no violad a h diffrn faciliis. Sa ransiions ha drmin whn aircraf nr (n) and xi (x) h sa of rquiring diffrn yps of mainnanc. For PM ˆ f ˆ f f R R x and ˆ f f ˆ f P ˆ n R x R c y rack h flying a a 1 a hours sinc h las PM. Th consrains yˆ Rˆ / c n f p a a a 1 a a a 1 and p work oghr o allow an aircraf o bgin PM whn ˆ f P c R c whr Rˆ c c ( y 1) hn f P f n ˆ a a f c is h minim fligh duraion immdialy bfor nring PM. Th consrains ak a similar form for nding PM and n bginning and nding RI. Ohr consrains of h form ˆ ˆx S S y S nsur aircraf a a 1 a 1 a 1 can only b in allowabl sas a allowabl ims such as whn hy bgin and nd mainnanc. A significan faur of h formulaion is h modl of UM. Thr rprsnaiv UM consrains ar: Rˆ Rˆ Wˆ x x a A T plm. flm a a 1 a 1 a a Rˆ c S a A T cfu n a a Rˆ c ( M c ) S a A T cfu cfu a a a W Th firs s of consrains racks h amoun of UM rquird for ach aircraf ach day whr 1 rprsns h nw UM MMH addd o an aircraf h prvious day. This is gnrad from a probabiliy disribuion and is hus xognous informaion o h modl. Th scond and hird ss of consrains prvn an aircraf from bginning UM if R cfu c. Th hird s of consrains us a big-m paramr in par bcaus R may b vry larg givn h righ-aild disribuion usd o gnra h valus of W alhough w hav drivd a im and aircraf-dpndn igh bound. For an aircraf o nd UM Ra rquirmns ar nforcd wih consrains similar o hos abov. 4. INDICATIVE RESULTS 4.1. Policis and inpus for compuaional xprimns c ; hs To dmonsra h uiliy of h modl w considr a drminisic look-ahad policy modl (Powll 2014) which has h form X ( S ) arg min x H ' C( S x ) C( S ' x ') ' 1 whr h indx indicas a bas-modl variabl (hs ar implmnd a h nd of day ) and ' indicas a look-ahad variabl (hs ar discardd a h nd of day ). Look-ahad policis opimiz ovr a im horizon (... H). Th modl implmns h dcision for h firs day hn solvs again ovr h im horizon and so on. Drminisic look-ahad policis us a poin sima o forcas nw UM vns. W ass no nw vns in h forcas of UM sinc his prformd wll and had improvd nrical sabiliy du o smallr big-m valus in comparison wih using a running man during iniial sing. W considr a squadron of 12 aircraf opraing ovr a 180 day priod. Aircraf can fly up o 6 hours pr p P day. Aircraf mus bgin PM bwn c 180 and c 220 flying hours (inclusiv) and rquir 300 MMH o rurn o srvicabiliy. RI ar rquird vry 7 days for 10 MMH. Thr is a oal of 150 MMH availabl ach o boh FLM and PLM wih a maxim of 50 MMH pr aircraf pr day and a capaciy of 3 mainnanc lins ach 4.2.for FLM and PLM. UM im bwn failurs ar drawn from a lognormal disribuion wih 3and 1 whil h im o rpair lognormal paramr valus ar 1 and 1.25. cfu Aircraf mus undrak UM whn c 20 and rurn o srvicabiliy whn c 5. In our conribuion 1396
Lookr al. Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc plm plm flm flm funcion w s 0.01 0.5( cmin / cmax ) 0.5( cmin / cmax ) and w f = 10. W iniialis h squadron wih 7 aircraf srvicabl 3 in phasd mainnanc and 2 in rgular inspcions wih h aircraf saggrd in ach sa.g. h 9 aircraf no in PM ar all sparad by 20 flying hours. W considr wo paramrs for our policy sarch: h im horizon H (from 1 o 14 days) and a discoun facor (ranging from 0.8 o 1). W wish o drmin which combinaion of im horizon and discoun facor produc h maxim xpcd flying hours as shown in h objciv funcion and implid by h choic of w f = 10. W us h givn valus of H and discriz h discoun facor by sps of 0.05 giving a oal of 70 dsign poins for h xprimns which wr conducd on a compur wih an Inl Xon 2.6 GHz CPU wih 64 GB of RAM running Ubunu 16.04LTS. Th modl was codd in Pyomo 5.0.1 and h solvr was Gurobi 7.5.0. Using 24 cors all 96 runs of h 14 day rolling horizon cas aks approximaly 90 minus o solv wih an opimaliy gap of 1% (for ach day); cass wih a smallr discoun facor run fasr han hos wih a largr valu. W prformd 96 runs a ach dsign poin as his is a Figur 1. Sampl oupu from a singl insanc for a singl dsign poin showing (a) h acculad flying hours sinc h las PM and (b) h rmaining acculad UM MMH mulipl of h 24 cors and rsuls in h widhs of h 95% confidnc inrvals bing lss han 5% rlaiv o h man. Figur 1 provids indicaiv schduls from a singl run (days 15-29) for on dsign poin o dmonsra how h modl works in pracic. Th abl on h lf (a) shows h valus for R ˆ f whil h abl on h righ (b) is for R. Grn rprsns a srvicabl aircraf blu is an aircraf in PLM orang is an aircraf in FLM and rd is an idl aircraf i.. nihr flying nor bing mainaind. Givn h objciv aircraf gnrally fly a h maxim ra. Figur 1(a) shows how aircraf Tabl 1. A smary of rsuls for h op 6 combinaions of im bgin PM a diffrn valus of horizon and discoun facor and h myopic policy (H = 0). ˆ f R : aircraf A10 afr 218 flying hours; A9 a 216 flying hours. RI occur on A4 on day 17 A3 on day 18 c. Th impac of UM is sn in Figur 1(b): aircraf A8 has 1763 MMH of UM rmaining a h nd of day 16 whn prviously srvicabl; and aircraf A10 incurs a las 573 MMH during a phasd srvic on day 21. Th UM consrains ar clarly vidn for aircraf A1: on day 16 i has 64 MMH of UM Tim horizon Discoun facor Flying hours (man) Flying hours confidnc inrval widh (%) 11 0.90 7355 3.38 9 0.95 7325 3.57 13 0.90 7321 3.04 14 0.80 7306 3.26 7 0.90 7303 4.05 14 0.90 7303 3.69 0 NA 3934 6.28 (576 runs) rmaining a h nd of h day hn a furhr 100 MMH is rmaining on day 19. Aircraf wih larg rmaining UM MMH ar givn a lowr mainnanc prioriy compard wih aircraf rquiring srvics ha can b compld mor quickly hus gnraing mor flying hours; his follows from h form of h conribuion funcion. 1397
Lookr al. Opimal policis for aircraf fl managmn in h prsnc of unschduld mainnanc Tabl 1 shows h 6 bs man oal flying hours rsuls oghr wih h myopic policy. Obsrv ha h bs cas wih a im horizon of 11 and ha wih a horizon of 7 ar no significanly diffrn; his indicas h ponial for savings in compuaion im. In fac h op 6 rsuls ar no significanly diffrn from ach ohr. Whil longr im horizons will produc largr opimal objciv valus for a givn paramr s and random daa simulaion rsuls may no significanly diffr whn compard wih shorr im horizons du o random variaion. Addiional saisical analysis indicas h lngh of h im horizon has a grar influnc on h oal flying hours han h discoun facor. 5. DISCUSSION AND FUTURE WORK W hav dvlopd a nw fl managmn modl for miliary aircraf ha xplicily includs random UM. Th modl also includs aircraf flying and schduld mainnanc and can b usd o sarch for policis ha achiv spcifid objcivs. Incorporaing UM givs his modl much grar fidliy han drminisic modls in h highly dynamic and uncrain nvironmn of miliary aircraf fl managmn. Fuur work will xplor h us of addiional policis such as sochasic look-ahad policis (Powll 2014) o drmin which policis bs m fl objcivs. I will also includ mor comprhnsiv ss of policy paramrs such as longr im horizons and rprsning h discoun facor as a coninuous variabl. Such xnsions may warran a mor horough ramn using simulaion xprimnal dsign chniqus. ACKNOWLEDGMENTS This rsarch has bn don undr a Collaboraiv Projc Agrmn bwn DST and Dakin Univrsiy. REFERENCES Cho P. (2011). Opimal Schduling of Fighr Aircraf Mainnanc. M. Sc. Thsis Massachuss Insiu of Tchnology. Gavranis A. and Kozanidis G. (2015). An xac soluion algorihm for maximising h fl availabiliy of a uni of aircraf subjc o fligh and mainnanc rquirmns. Europan Journal of Opraional Rsarch 242 631-643. Hahn R. A. and Nwman A. M. (2008). Schduling Unid Sas Coas Guard hlicopr dploymn and mainnanc a Clarwar Air Saion Florida. Compurs & Opraions Rsarch 35 1829-1843. Kozanidis G. Gavranis A. & Kosarlou E. (2012). Mixd Ingr Las Squars Opimizaion for Fligh and Mainnanc Planning of Mission Aircraf. Naval Rsarch Logisics 59 212-229. Marlow D. O. and Dll R. (2017). Opimal shor-rm miliary aircraf fl planning. Submid o Journal of Applid Opraions Rsarch. Marlow D. O. and Novak A. (2013). Using discr-vn simulaion o prdic h siz of a fl of naval comba hlicoprs. In Pianadosi J. Andrssn R.S. and Boland J. (Eds) MODSIM2013 20h Inrnaional Congrss on Modlling and Simulaion Adlaid: Modlling and Simulaion Sociy of Ausralia and Nw Zaland 2506 2512. Maila V. & Virann K. (2011). Schduling Fighr Aircraf Mainnanc wih Rinforcmn Larning. In S. Jain R. R. Crasy J. Himmlspach K. P. Whi and M. Fu (Eds.) Procdings of h 2011 Winr Simulaion Confrnc. Phonix: IEEE 2540-2551. Maila V. & Virann K. (2014). Mainnanc schduling of a fl of fighr aircraf hrough muliobjciv simulaion-opimizaion. Simulaion: Transacions of h Sociy of Modling and Simulaion Inrnaional 90(9) 1023-1040. Maila V. Virann K. and Raivio T. (2008). Improving mainnanc dcision-making in h Finnish Air Forc hrough simulaion. Inrfacs 38(3) 187-201. Pippin B. (1998). Allocaing Fligh Hours o Army Hlicoprs. M. Sc. Thsis Naval Posgradua School. Powll W. (2011). Approxima Dynamic Programming: Solving h Curs of Dimnsionaliy. Wily Sris in Probabiliy and Saisics 2 nd diion. Wily Hobokn Nw Jrsy. Powll W. (2014). Claring h jungl of sochasic opimizaion. Bridging Daa and Dcisions INFORMS Tuorials on Opraions Rsarch 109-137. 1398