Available online at ScienceDirect. Conference on Systems Engineering Research (CSER 2014) Zhemei Fang*, Daniel DeLaurentis

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1 Available online a ScienceDirec Procedia Compuer Science 28 ( 214 ) Conference on Sysems Engineering Research (CSER 214) Eds.: Azad M. Madni, Universiy of Souhern California; Barry Boehm, Universiy of Souhern California; Michael Sievers, Je Propulsion Laboraory; Marilee Wheaon, The Aerospace Corporaion Redondo Beach, CA, March 21-22, 214 Dynamic Planning of Sysem of Sysems Archiecure Evoluion Zhemei Fang*, Daniel DeLaurenis Purdue Unicersiy, 71 W. Sadium Avenue.,Wes Lafayee, IN 4797, USA Absrac The dynamic planning and developmen of a large collecion of sysems or a Sysem of Sysems (SoS) pose significan programmaic challenges due o he complex ineracions ha exis beween i consiuen sysems. Decisions o add, remove, or reconsiue connecions beween sysems can resul in repercussive failures across operaional and developmenal dimensions of an SoS. The work conduced in his research is par of a larger body of work funded by he DoD Sysems Engineering Research Cener (SERC) owards he developmen of an Analyic Workbench. This paper in paricular develops a ool ha adop an operaions research-based perspecive o SoS level planning based on merics of cos, performance, schedule and risk. Specifically, our work employs an Approximae Dynamic Programming approach ha is well suied o address issues of compuaional racabiliy of he resuling dynamic planning opimizaion problem. This approach allows for idenificaion of near-opimal muli-sage decisions in evolving SoS archiecures. A Naval Warfare Scenario SoS example problem illusraes applicaion of he mehod. 214 The Auhors. Published by by Elsevier B.V. B.V. Open access under CC BY-NC-ND license. Selecion and peer-review under responsibiliy of of he he Universiy of of Souhern California. Keywords : Sysem of Sysems Archiecure Evoluion; Approximae Dynamic Programming; Planning; Uncerainy 1. Inroducion US Deparmen of Defense (DoD) has recognized he prevalence of a Sysem of Sysems view o he acquisiion and developmen of miliary asse 1 ; his means ha SoS capabiliies being sough are a direc consequence of he * Corresponding auhor. Tel.: ; address: fang59@purdue.edu The Auhors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selecion and peer-review under responsibiliy of he Universiy of Souhern California. doi: 1.116/j.procs

2 45 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) ineracive effec of heir consiuen sysems. These consiuen sysems possess operaional and managerial independence, ye inerac on various levels o give rise o an overarching SoS level capabiliy. Decisions o suppor he developmen of hese monolihic eniies have required acquisiions using sysems engineering based policies ha can beer accoun for he complexiies associaed wih SoS archiecures. To his end, Defense Acquisiion Guidebook (DAG) 2 has been developed o aid he undersanding and implemenaion of DoD acquisiion pracices including evoluionary acquisiion sraegies ha are he norm for SoS capabiliy evoluion. Consisen wih DAG, Sysems Engineering Guide for Sysems of Sysems (SoS SE) 1 examines he SoS challenges and provides a Trapeze model o give a good concepual view of he SoS SE core elemen, heir inerrelaionships and SoS decision-making arifac. Dahmann 3 unwinds he rapeze model o a more familiar and inuiive ime-sequenced Wave model and idenifies informaion criical o decision making in SoS evoluion. Fig.1 illusraes he wave model and original SoS SE core elemen. Fig.1. Wave Model and Relaed SoS SE Core Elemen 3 The sysemaic procedures as shown in Fig.1 provide guidance o SoS praciioners o make decisions properly; in addiion o he logical procedures, an analyic soluion framework o objecively quanify he sae and oucome of consequen acions o evolve an SoS archiecure is required by SoS praciioners. Guided by he wave model, Plan SoS Updae and Implemen SoS Updae are core par o drive an SoS forward, where key decision poin for SoS archiecure evoluion are mainly locaed. Accordingly, acions may involve a sequence of decisions ha include adding new sysems, reiring old sysems, upgrading sysem, ec for Plan SoS Updae ha could provide policy makers decision se for achieving opimal or near-opimal SoS capabiliy over a ime period. Operaional decisions for Implemen SoS Updae migh be inegraed meanwhile o provide promp feedbacks o developers. Unlike radiional producion, invesmen or supply chain planning problem a sysem level, he dynamic planning in an SoS exhibi a muliude of disinguishing feaures ha need o be carefully addressed. Typical quesions could include: how o deal wih he ineracions beween decisions from muliple independen organizaions, how o deal wih he diverse ime scales occurring in an SoS (such as invesmen decisions every five year versus operaional deploymen every few monhs), how o deal wih he complexiy resuling from he sheer number of uncerain variables involved, and so forh. The sequenial developmen process and he objecive of maximizing he overall capabiliy for a ime period makes dynamic programming a naural choice o address he problem. However, he characerisics of an SoS such as he large number of sysems ha may be involved, muli scale decisions and significan uncerainies lead o sae, decision and sample explosion respecively, which challenges he use of dynamic programming. Approximae Dynamic Programming (ADP) is an umbrella covering various mehods and echniques, aiming o solve he hree curses of dimensionaliy primarily by approximaing he fuure value funcions. Thus his paper employs ADP such ha he SoS archiecure evoluion process can be formulaed ino a dynamic planning problem and meanwhile complexiy from muli ime scale decisions and uncerainies can be addressed.

3 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) Lieraure Review Researchers have come up wih differen frameworks and mehods o ackle archiecing and evoluion of sysems and SoS. Ross 4 proposes an Epoch-Era Analysis for sysem design comparison and selecion o deliver susained value o sakeholders in he face of a rapidly changing world. Epoch-Era Analysis uses naural value-cenric ime scales, where Epoch refers o a period wih a fixed conex, characerized by saic consrain, available design concep, available echnology, and ariculaed aribues while an Era is generaed by sringing ogeher sequences of epochs given he likelihood of swiching beween given epochs and he duraions of each epoch. A muliude of exended work based on Epoch-Era Analysis has been conduced and he primary focus is he valuaion of changeabiliy under he framework 5,6. The curren applicaion of changeabiliy focuses on complex sysem design using qualiaive indicaors while he applicabiliy o SoS has no been fully addressed. Overall, Epoch-Era Analysis provides a concepual framework ha needs o be combined wih oher analyical echniques such as opions heory o guide he evoluion. Real Opions Analysis (ROA) is an approach ha values flexibiliy when cerain physical decision opions are embedded o cope wih fuure uncerainies 7,8,9. Simply pu, a real opion gives he righ (bu no he obligaion) o underake some business iniiaives such as abandoning, deferring, expanding projec and so forh. De Weck employs ROA o invesigae he benefi of he saged deploymen of communicaions saellie consellaions in low earh orbi under demand uncerainy 1, which shows he usefulness of ROA for iniial archiecure selecion embedded fuure opions. However, he discree number of decision ree opions grows significanly wih he number of available opions involved. For an SoS, his number is a combinaorial arifac of he number of curren and ye-o-be inroduced candidae sysems involved and hus is prone o he curse of dimensionaliy. Porfolio heory has been applied o decision-making in an SoS environmen by reaing he collecion of exising and poenial fuure sysems as a porfolio of asse sysems, which combine o deliver a desired SoS level goal. The mos uilized form of porfolio heory involves he applicaion of mahemaical programming mehods in idenifying opimal collecions of invesmen asse ha balance reward agains risk, given an invesor s specific olerance for risk. Prior work in SoS archiecural analysis has uilized robus porfolio opimizaion echniques in addressing he acquisiion of consiuen sysems in a sysems of sysems 11. Oher work employs muliple objecive value analysis, mahemaical opimizaion, cos benefi analysis, mean-variance approach and so forh o suppor porfolio decision analysis 12. However, hese classes of porfolio mehods do no direcly ranslae o muli-sage porfolio problems as readily for SoS archiecure evoluion. Dynamic programming was proposed as one of he mos promising mehods o formulae he SoS managemen problem by Maier 13. However, due o he large number of sysems and inheren uncerainies involved in an SoS, dynamic programming suffers from he compuaional complexiy, also known as he curse of dimensionaliy. On he oher hand, approximae dynamic programming presen a powerful modeling and algorihmic sraegy ha can address a wide range of opimizaion problems ha involve making decisions sequenially in he presence of differen ypes of uncerainies 14,15. I employs a variey of approximaion echniques in addressing issues of compuaional racabiliy due o he curse of dimensionaliy. Berekas 16 applied neural nework concep o approximae he value funcion and named Neuro-Dynamic Programming (NDP), which is he same as ADP. Powell 17 developed approximaion sraegies based on pos-decision sae variables which avoid compuing he expecaion of uncerainies. Wide applicaions of ADP exis in lieraure on dynamic resource allocaion problems. Powell e al have applied ADP echniques o real world problem including miliary airlif operaions under uncerainy 18, flee managemen involving railers, conainers, locomoives and so on 19, R&D porfolio opimizaion for solid oxide fuel cells problem 2, energy resource allocaion problem wih hourly ime incremen over an enire year or several decades 15, ec. Oher research work such as modeling global climae policy under decision-dependen uncerainy 21, and muli-sage invesmen managemen 22, ec, has also been invesigaed using ADP. The prior cied applicaions of ADP have shown i poenial in addressing issues of compuaional racabiliy and sequenial decision-making under condiions of uncerainy. The decision variables involved reflec he same kind of resource allocaion decisions ha are exercised in SoS archiecural decision making; however, none of hem have aken ino accoun he SoS characerisics such as individual sysems making heir own decisions and complex ineracions among myriad heerogeneous sysems, ec.

4 452 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) Technical Approach Fig.2. Overall Framework for he Sequenial Decisions This paper formulaes he process of SoS archiecure evoluion shown in Fig.1 as a dynamic programming model. Fig.2 demonsraes a hierarchical framework wih muliple ime-scales o solve he SoS archiecure evoluion as a dynamic planning problem. The objecive is o maximize he overall SoS capabiliies subjec o a se of resource consrain like budge and manpower, over a ime horizon, under uncerainy. The sequenial decision variables are in he form of addiion, removal and coninuaion of sysems. The approach ranslaes he hierarchical and coupled naure of inerconneced sysems wihin an SoS ino he language of mahemaical programming ha equivalenly describes characerisics of he problem wihin he conex of an opimizaion problem. Once we formulae he problem ino a dynamic programming model, various approximaion sraegies including aggregaion, parameric model and non-parameric model can be applied o he resuling muli-sage problem o address complexiy and compuaional racabiliy. As illusraed in Fig.2, he overarching objecive is o maximize a given SoS level capabiliy index over a ime horizon. The capabiliy index can be ranslaed ino he number of hrea being engaged, he number of survival aircraf or ships afer being aacked, ec. High level sraegic decisions address long ime scale decisions such as invesmen or acquisiion of new aircraf, ships, saellies, ec which all paricipae in a paricular SoS. Muliple years effor is needed o obain he final produc afer curren decisions. Ambiious schedule for developing all new sysems a he beginning is no wise, as i leads o cos and schedule overruns and performance degradaion. Sequenial decisions are preferable as hey offer he chance of evaluaing and learning he curren sae such as echnology mauriy, budge siuaion and so forh from he environmen. Low level decisions can eiher represen operaional decisions or decisions wihin consiuen sysems like scheduling. In his paper, operaional decisions ac as low level decisions wih shor ime scale. For insance, during a miliary deploymen, decision makers have o sequenially decide when o pu sysems in service, when o pu sysems in mainenance and when o keep sysems idle, whereby opimal performance migh be achieved under random incoming aacks. The high level sraegic decisions provide a resource pool for he low-level operaional decisions; new invesed sysems can become available afer a few years, by which ime operaional decision-makers will have addiional choices and more advanced sysems a heir disposal. Low level shor ime scale decisions are required for wo reasons: one is o saisfy shor ime requiremen; he oher, is o provide more accurae feedback and new informaion o high level decision-makers o faciliae learning. Large errors occur if new informaion is couned and learn afer a few years

5 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) wihou considering effec of shor ime scale decisions. Moreover, a variey of sources of uncerainies or exogenous informaion exis in SoS, such as naional prioriy, echnology, budge, marke, climae change, ec, all of which have compleely differen ime scales. For example, naional prioriy may have major changes every five year while budge siuaion migh change every year and marke demand migh vary every week. Therefore, his framework aims o provide policy makers a sequence of archiecure alernaives a differen sages and ime scales (ime inerval beween decisions), given he individual sysem capabiliies and resource consrain. The incorporaion of muli ime scales leads o a quick increase in he number of decision variables involved in an SoS. If we consider a 1 year long SoS program as an example, annual sraegic decisions and monhly operaional decisions will give us 12 decision variables. Besides, he large number of sysems included in an SoS along wih he heerogeneiy of sysems, generaes a large number of archiecure alernaives, no o menion he effec of complex uncerainies. Hence, all hese pose significan challenges o compuaional efficiency. Approximae dynamic programming framework provides a pah o address such ype of issues hrough approximaion of he value funcions and many oher echniques, as depiced in he nex secion. 4. Approximae Dynamic Programming The beauy of ADP lies in he capabiliy of generaing decisions based on an approximaion of expeced fuure value and corresponding learning and updaing from resulan new informaion. I alleviaes he pain of collecing complee informaion and building a perfec model for opimizing a complex sysem of sysems. Basically, he compuaional cos only sems from he producion of ieraion number and sage number, which gives grea efficiency when he problem grows large, alhough he efficiency needs o sacrifice cerain amoun of accuracy of resul. In his paper, wo ADP sraegies are employed: linear value funcion approximaion as a special form of parameric model and he concep of pos-decision sae variables, as used in lieraure 17. Classical dynamic programming recursively compues he Bellman equaion which is he essence of dynamic programming as following: V( S ) max ( C ( S, x ) E{ V ( S ) S}) (1) 1 1 x X S represen sae variables, where x represen decision variables, C means curren conribuion, is discoun facor and V 1( S 1) means expeced value of being in sae S 1. Exac fuure value is almos impossible o obain, approximaed value V 1 is used insead. A generic srucure for he value funcion approximaions is V ( S ) ( S ) (2) f f f F where { ( S ): f F} are ofen referred o as feaures ha capure he imporan characerisics of he resource f sae vecor from he perspecive of capuring he oal expeced conribuion in he fuure. f represen adjusing parameers ha allow us o obain differen value funcion approximaions. A variey of mehods exis o calculae and ( S ). Linear and piece-wise linear value funcion approximaions are wo ypical insances. f f To avoid he calculaion of expecaion, pos-decision sae variables are inroduced, by which Bellman equaion can be wrien as: x x x M, x x V ( S ) E{max C ( S, x ) V ( S ( S, x )) S } (3) x X x where S 1 represen pos-decision sae vecor. In his equaion, expecaion can be dropped by using a sample realizaion of he uncerainies W ( w ); hen he equaion urns o he following form: x x x M, x x V ( S ) max C ( S, x ) V ( S ( S, W( w), x )) S (4) x X Given a paricular realizaion of ( ) W w, he above equaion becomes a deerminisic opimizaion problem, which solves he curse arising from he size of decision space.

6 454 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) Illusraive Example and Resul The Lioral Comba Ship (LCS) is a relaively new naval sysem ha, ogeher wih i relaed componen, may be viewed as a naval warfare SoS. LCSs are oufied wih hree differen mission packages - surface warfare (SUW), ani-submarine warfare (ASW), and mine warfare (MIW), aimed a counering mines, small boa and submarines in lioral waers 23. The SUW module ha is designed o deec and engage muliple surface conac in a lioral environmen consis of LCS, UAV, and MH-6R (helicoper). We demonsrae he applicabiliy of our approach by applying i o a simplified SUW module. A capabiliy index can be convered from percenage of sysems surviving an aack and for he sake of simpliciy, noional numbers based on exper judgmen are used. In his conex, we assume ha a sraegic level, his is a hree-year program and a sysem can be compleely developed in one year. Decision makers wish o be aware of which archiecure from differen combinaions of LCS, UAV and MH-6R should be developed a he beginning of each year. Once sysems are available o ener he operaional domain, we assume he deploymen of hese sysems are seasonal and decision makers wish o know wheher o pu exising sysems in service or ou of service (like mainenance) o preven uncerain aacks. The feedback from he low level decisions can influence he sraegic decisions for nex year. This process reflec back o he original flow in Wave model of plan SoS updae and implemen SoS updae. A pre-requisie o formulaing he problem ino a mahemaical programming forma is o idenify he basic elemen of dynamic programming: sae variable, decision variable, ransiion funcion, exogenous informaion (uncerainy) and objecive funcion. The objecive is o maximize he expeced sum of SoS capabiliy a operaional level afer being aacked during each sage, wih he consrain of budge on he invesmen of new sysems a he beginning of each year. I is assumed ha he capabiliy of each sysem is addiive owards obaining SoS level performance. Low level decisions are assumed o be binary. Accordingly, he formulaion can be wrien as: 3 4 p obj :max E( c ( R y )(1 wˆ ) 1 s 1 in,,, 1 s s : c x B ( 1..3) and y R ( 1..3, s 1..4) ou,, p where c represen a row vecor of noional index of individual sysem (LCS, UAV, MH-6R) capabiliy and is assumed o be consan over ime. wˆ, s 1 as uncerainy coming from ime s o s 1, refers o a column vecor of binary resul from he producion of probabiliy of hrea occurring and probabiliy of successful aack. c s denoes developmen cos of each sysem while B means budge limi a each year. Sae variable R represen new a vecor of numbers of sysems wih differen aribues a season s of year, specifically, R consis of R, in ou new in ou R, and R where R is for new sysems under developmen, R is for sysems in service while R is for sysems ou of service. Decision variable x means developing sysems a he beginning of each year (for he ease of represenaion, i means he new available sysems ha are developed one year ago) while y, means puing available sysems in service. Transiion funcions can be formulaed as follows: ou ou new sysems available : R, 1 R 1, s 4 x, 1 (6) in in ou ou pu in service : R ( R y )(1 wˆ ); R R y, 1,,, 1, 1,, We assume he iniial number of available sysems in he archiecure is zero, hus he resource pool for low level decisions wihin one year is direcly from iniial sraegic level decisions. Thus he problem can be furher simplified. 12 p obj : max E( c ( R y )(1 wˆ )) s 1 in s s s 1 4( 1) 4 4( 1) 4 s s s s 4( 1) 1 s 4( 1) 1 s : c y B ( 1,2,3) where x y ( 1,2,3) Currenly value funcion approximaion V( S ) is assumed as a linear funcion of he number of resources, by which we assume o be able o capure he expeced fuure conribuion. More sophisicaed value funcion approximaion will be sudied in fuure work. In order o validae he resul from ADP, a regular ineger (5) (7)

7 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) programming solved by Gurobi is employed o obain he opimal soluion for comparison. To easily compare he resul from ADP and opimal soluion, incoming aacks are assumed o be deerminisic and known as a prior. Under his experimen seing, resul can be obained as displayed in Fig.3 and Fig.4. Noe ha he primary reason for employing ADP relies on i poenials of compuaional scalabiliy o solve problems wih large number of saes and uncerainies involved where opimal values are usually difficul o compue. Since his example is no a large problem, he advanages of ADP are no obviously demonsraed. Resul in Fig.3 and Fig.4 primarily aim o illusrae ha reasonable objecive can be obained by ADP in small examples and i is validaed o furher apply o large problems. Objecive Value ADP Ieraion Number Opimal Value Fig.3. ADP Objecive Value and Opimal Objecive Value Number of New Sysems LCS UAV MH-6R Time Sep (Year) Number of LCS in Service Time Sep (Season) Number of UAV in Seervice Time Sep (Season) Number of MH-6R in Service Time Sep (Season) Fig.4. Sraegic Level Decisions and Operaional Level Decisions Fig.3 shows he comparison beween objecive value obained by ADP and opimal objecive value. I indicaes ha performance sabilizes afer around 5 raining ieraions of value funcion approximaions. The ADP objecive value is 6% lower han he opimal value and he accepance of his sub-opimal resul is dependen on he decision makers preference of compuaional efficiency and accuracy of he resul. Compuaional cos of ADP linearly scales wih he produc of ieraion number and ime sages while regular linear or ineger programming exponenially scales when sysem saes and uncerainy increase. Overall ADP provides subopimal decision se wih a reasonable range; when sysems involved becomes large, ime scale grows and uncerainies are presen, ADP exhibi a clear advanage in compuaional efficiency. As shown in Fig.4, sequenial decisions of archiecure alernaives a sraegic and operaion level can be provided for decision makers and i gives policy makers a global sense of decisions and resuling effec owards he SoS. 6. Conclusion and Fuure Work This paper addresses decision-making in he dynamic planning of an SoS archiecure using approximae dynamic programming echniques. A noional Naval Warfare SoS is used o illusrae he applicabiliy of he mehod.

8 456 Zhemei Fang and Daniel DeLaurenis / Procedia Compuer Science 28 ( 214 ) Deerminisic assumpions are made o compare he soluion of he ADP algorihm o he opimal soluion. The resul indicae a sequence of archiecure alernaives over ime for SoS praciioners while a small amoun of difference exis beween he objecive value from ADP algorihm and opimal objecive value. Fuure work will explore applicaion of algorihmic advances in he ADP formulaion so as o enable efficien incorporaion of forward sae informaion, learning poenial and quanificaion of uncerainies relaed o he value in being in paricular saes (archiecures). Addiionally, fuure work will explore compuaional efficiency using a more realisic, large scale SoS concep problem. Acknowledgemen This maerial is based upon work suppored, in whole or in par, by he U.S. Deparmen of Defense hrough he Sysems Engineering Research Cener (SERC) under Conrac H D-171. SERC is a federally funded Universiy Affiliaed Research Cener managed by Sevens Insiue of Technology. References 1. OUSD(AT&L). Sysems Engineering Guide for Sysem of Sysems. Washingon, D.C.: Penagon:; Augus DoD. Defense Acquisiion Guidebook. ; Dahmann J, Rebovich G, Lowry R, ec. An Implemeners' View of Sysems Engineering for Sysems of Sysems. In ; 21. p. pp Ross A, Donna R. Using Naural Value-Cenric Time Scales for Concepualizing Sysem Timelines hrough Epoch-Era Analysis. In ; Jun Fizgerald M, Ross A. Miigaing Conexual Uncerainies wih Valuable Changeabiliy Analysis in he Muli-Epoch Domain. In Sysems Conference (SysCon), 212 IEEE Inernaional; March 212. p Fizgerald M, Ross A. Susaining Lifecycle Value: Valuable Changeabiliy Analysis wih Era Simulaion. In Sysems Conference (SymCon), 212 IEEE Inernaional; March 212. p de Neufville R. Real Opions: Dealing wih Uncerainy in Sysems Planning and Design. Inegraed Assessmen. 23; Vol.4(No.1): p Chaize M. Enhancing he Economics of Saellie Consellaions via Saged Deploymen and Orbial Reconfiguraion. MIT; Mikaelian T. An Inegraed Real Opions Framework for Model-based Idenificaion and Valuaion of Opions under Uncerainy. MIT; Jun Weck Od, Neufville Rd, Chaize M. Saged Deploymen of Communicaions Saellie Conellaions in Low Earh Orbi. Journal of Aerospace Compuing, Informaion, and Communicaion. 24 March; Vol.1: p. pp Davendralingam N. Formulaion and Demonsraion of a Robus Mean Variance Opimizaion Approach for Concurren Airline Nework and Aircraf Design.; Burk R, Parnell G. Porfolio Decision Analysis: Lessons from Miliary Applicaions. In Salo A, Keisler J, Moron A. Porfolio Decision Analysis: Improved Mehods for Resource Allocaion.; 211. p. pp Maier M. Research Challenges for Sysems-of-Sysems.; Simao H, Day J, ec. An Approximae Dynamic Programming Algorihm for Large-Scale Flee Managemen: A Case Applicaion. Transporaion Science. 28;: p. pp Powell W, George A, Simao H, ec. SMART: A Sochasic Muliscale Model for he Analysis of Energy Resources, Technology, and Policy. Informs Jounal on Compuing. 211;: p. pp Berekas D, Tsiikilis J. Neuro-Dynamic Programming: Ahena Scienific, Belmon, MA; Powell W. Approximae Dynamic Programming: Solving he Curses of Dimensionaliy. 2nd ed.; Powell W, Ayari B, Berger J, ec. The Effec of Robus Decisions on he Cos of Uncerainy in Miliary Airlif Operaions. ACM Transacions on Modeling and Compuer Simulaion. 21; Vol.9(No.4): p. pp Powell W, Topaloglu H. Flee Managemen. ; Hannah L, Powell W, Sewar J. One-Sage R&D Porfolio Opimizaion wih an Applicaion o Solid Oxide Fuel Cells. Energy Sysem. 21;: p. pp Webser M, Sanen N, Parpas P. An Approximae Dynamic Programming Framework for Modeling Global Climae Policy under Decision- Dependen Uncerainy. ; Keles P, Harman J. Evaluaing Porfolios of Muli-sage Invesmen Projec wih Approximae Dynamic Programming. ; Ronald O. Navy Lioral Comba Ship (LCS) Program: Background and Issues for Congress. Congress Research Service; 213.

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