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Graduate Research Award program Application: 2010-2011 Page 2 of 10

Applicant's name:Last First Middle

Date: 10 Mav 2010

A completed application checklist and application package cover sheet

Personal information of applicant

Qualifications of applicant

Research project proposal

Reference letter #1

Reference letter #2

Research advisor form

Officialtranscripts from all colleges and universities attended

Please list all institutions from which officialtranscripts have been requested. Transcripts may beincluded in the submission packet if properly sealed from the registrar.

lf official transcripts will be sent separately, please indicate that they have been ordered andinclude an unofficial copy of each of your transcripts in the application package.

Writing Sample

(The writing sample should be submitted with the application. An appropriate writingsample might be a previous publication, a paper written for a class assignment, aresearch project report, or similar example of professional writing. lf papers were co-authored, the role of the applicant must be clearly described. Writing samples may notbe longer than 25 pages.)

V]

Graduate Research Award Program Application: 2010-2011 Page 3 of 10

Administered By:Airport Cooperative Research ProgramTransportation Research Board, National Academies

APPLICATION FORM

GRADUATE RESEARCH PROGRAM ON PUBLIC.SECTOR AVIATION ISSUES

Sponsored By:Federal Aviation AdministrationU.S, Department of Transportation

PART I- PERSONAL INFORMATION OF APPLICANT

(Please Type)

1. Full legal name:

First Former name (if any)

2 Date of birth: 19 OCT 1973 place of birth: Washinqton, D,C.3. Citizenship: U.S.

4. Gender: IX ] Male [ ] Femate

5. Ethnicity (optional) :

[ ]American lndian orAlaskan Native: origin in any of the original peoples of North AmericaI X] Black. origin in any of the black raciat groups[ ] Hispanic: Mexican, Puerto Rican, Central or South America, or other Spanish culture or origin,

regardless of race[ ] Asian or Pacific lslander: origin in any of the original peoples of the Far East, Southeast Asia,

or the pacific lslands. lncludes China, Japan, Korea, the Philippine lslands, Samoa, and thelndian Subcontinent

[ ] White: origin in any of the original peoples of Europe, North Africa, or the Middle East

Mailing address: 2032 Woodcliff Drive. Smvrna, TN 37167

Permanent address: 2032 Woodcliff Drive. Smvrna. TN 37167

Telephone numbers - Mailing: _615-410-7149- permanent: _423-sos-5967

Email address: _quentin.c. noreiga@vanderbilt.edu

College or University currently enrolled at: Vanderbilt. UniversityMajor Field: _Systems EngineeringDegree objective: [ ] Maste/s [X ] DoctorateExpected month and year of graduation: _ MAY 2012

Names of two people from whom you are requesting reference letters.

a. Dr. Mark P. McDonald

b. Dr. Sankaran Mahadevan

Name and title of faculty research advisor for this project:

Dr. Mark P. McDonald, Assistant Professor,Civil & Environmental Engineering, Vanderbilt University

Middle

6.

7.

8.

9.

10

11.

12

Graduate Research Award Program Application: 2010-2011 Page 4 of 10

PART II_ QUALIFICATIONS OF APPLICANT

(Please Type)

13. Education: ln reverse chronologicalorder, list colleges or un¡versities attended.

Please explain any interruption(s) of schooling, i.e., military training, illness, etc.). Full time Army officer from JUN 1995 until beginning of current program in AUG 2008.

14. Professional Experience. ln reverse chronologicalorder, list professionalexperience, includingsummer and term-time work.

15. Awards, honors, and publications: List fellowships, scholarships, and other academic and/orprofessional positions, held since entering college or university,

College / University Location Major FieldDates

Attended GPA Degree Date degreeawarded/exoected

/anderbilt University \ashville,fN

3ystems:NG

JANlO-lurrent

).042

)hD\lAY 2012

r/anderbilt University \ashville,fN livil ENG

\UGO8.)EC09

/1.s. lEc 2009

)ld Dominion Universitytlorfolk,,/A

Jusiness\dmin

JANOS-VIAYOT

).2MBA

vlAY 2007

J.S. Military AcademyWest Point)

/Vest)oint, NY

vlechanical:NG

JUL91-,UN95

¿.03.S.

JUN 1995

Name of Emolover Location Dates Nature of WorkU.S. Army Afghanistan JUL 07- JUL 08 Operations Research

Systems Analvst (ORSA)U.S. Army Fort Monroe, VA ocT 04 - JUN 07 Operations Research

Svstems Analvst (ORSA)U.S. Armv Chattanooqa, TN JUN 02 - MAY 04 Recruitino CommanderU.S. Army Savannah, GA JUN OO - MAY 02 Ammunition Supply

Commander

Award, Honor, or publication Date(s) Description

fRB 2010 Poster Presentation JAN 2O1O

)roblem Decomposition Methods for Joint Airport{etwork Design, Aircraft Design, and Aircraftìoutinq (10-3561)

Graduate Research Award Program Application: 2010-2011 Page 5 of 10

16. Describe your career goals and how this research will contribute to achieving those goals (you

may add page(s) if necessary):

I am trained as an Operations Research Systems Analyst and as a logistics specialist bythe U.S. Army. My career goals are to return to the Department of Defense as a highly qualifiedTransportation Supernetwork analyst working on high level personnel and equipmentmoblilization issues, My research in multi-modal transportation supernetwork design andanalysis will be fundamental in achieving my career goals. Eventually, I also plan to teachSystems Engineering and the United States Military Academy, where I earned my undergraduatedegree.

I began work on a MS/PhD program in the department of civil engineering at VanderbiltUniversity in fall of 2008 after spending l2-months deployed to Bagram, Afghanistan as anImproved-Explosive-Device (IED) battle tracker. My previous work assisted maneuver units inroute and supply network planning which in part led me to pursue a Doctorate of Philosophy inSystems Engineering & Operations Research, focusing on commercial air and groundtransportation applications.

I am interested in finding solutions to the problem of managing congested transportationnetworks for inter-city travel. The American airspace is rapidly approaching gridlock, and nosingle solution to this problem is readily apparent. However, it becomes clearer that the solutionto this problem will involve some combination of expanding airport capacity, utilizing betteraircraft (including designing new aircraft that can help alleviate congestion problems) and betterlogistical models in fleet assignment planning, including high speed rail alternatives to airtransportation, and manipulating supply-demand economics. Currently, planning of projects tomanage airspace congestion is done in an isolated ad-hoc approach, partially because there is nosingle body with authority to integrate the planning process, but also partially because there is nocurrently existing methodology to integrate the various issues of this transportation systemsplanning problem into a planning model that can be used for decision support. Fudher, these

decisions are made under uncertainty regarding future demands on the transportation network.The development of a planning model that can account for all of these aspects of the problemwould have a large impact on transportation engineering because it can allow various planningagencies to examine the benefits of partnering to provide better solutions to managing gridlock.It would allow those advocating for high speed rail or airport capacity expansion projects toquantify the impacts of their projects at the level of the total transportation system.

My initial interest in ground transportation stemmed from being a weekend commuter onV/ashington D.C. subway system. As a child, my mother took me on Saturday morning trainrides as a weekend treat for good behavior and good grades. I often asked her why the trainstations were so close together, why the train routes crossed at certain stops, and who decidedwhen the train should travel? These childish curiosities, over time, developed into graduateinterests in problem solving methods, graph theory, network design, and optimal routing oftransportation networks. This past January (2010), I lost my mother after her prolonged fightwith diabetes and severe renal failure. My childhood memories of her combined with theeuphoria of my childhood train rides have further etched in stone my desire to pursue studies intransportation engineering.

I affirm that the information provided in this application is true and complete to the best of my knowledge.

Signature of the Applicant: ...

INOTE: This application is not complete without a signature.]

Date./2.-4f.7 zoto

APPLICATION FORM

GRADUATE RESEARCH AWARD PROGRAM ON PUBLIC.SECTOR AVIATION ISSUES

Graduate Research Award Program Applicat¡on: 2010-2011 Page 6 of 10

Administered By:Airport Cooperative Research ProgramTransportation Research Board, National Academies

Sponsored By:Federal Aviation AdministrationU.S. Department of Transportation

PART III -RESEARCH PROJECT PROPOSAL

Name: Noreiqa, Quentin, Carlyle Date: 10 May 2010Last First

Title of Research Project:

Middle

Approximate Stochastic Optimization Techniques for Transportation Supernetworksln 500 words or less, describe the proposed research project. lnclude project objectives,methodology, and expected outcomes. Also indicate how this research work could benefit theaviation community, and contribute to your career goals.

My plan of study seeks to develop foundational mathematical and computationalmethods to solve practical multi-modal transportation problems. Using optimization language,these problems are multidisciplinary, multi-stage, stochastic, mixed integer nonlinearprogramming problems (MINLP), Currently, no solution methods exist for these problems in themulti-modal transportation domain. ln my research, I will develop methods for solving theseproblems and apply them to transportation supernetworks. I will solve these networkoptimization problems as two stage stochastic programming problems, approximate dynamicprogramming problems, and multi-objective problems, and then apply the methods to atransportation supernetwork of combined air, road, and high speed rail.

Objective One: I will develop a methodology to solve a two-stage stochastic programwith recourse using surrogate modeling for faster and more efficient computing. A stochasticprogram contains inputs values that have a degree of uncertainty. The problem is two-stage inorder to incorporate decisions made at different time intervals. Decisions made during stageone become constraint values or input parameters to be used in stage two, with the stage-twodecision to be made after realizing an uncertain scenario. I have developed an eight-stepdecomposition process which begins by formulating a 'master' problem which optimizes stageone and stage two decisions together. Though conceptually simple, the master problem is

usually too computationally expensive to solve. This process is outlined in my 2010 TRB paper,

Problem Decomposition Methods for Joint Airport Network Design, Aircraft Design, and AircraftRouting.

Objective Two: I will apply the two-stage stochastic programming method developed in

objective one to the road, HSR, and air supernetwork in the California Corridor (Los Angeles to

San Francisco). I will seek to minimize the sum of capital costs and operational costs for thetotal transportation system. The objective function will include some combination of agencycosts, user costs, and environmental consequences.

Objective Three: I will develop an approximate dynamic programming (ADP)

methodology to develop a multi-stage theoretical approach. ADP leverages the power of statevariables, or variables descriptive of all the information that is needed to make a decision at aparticular stage. The optimization process works backwards, solving optimization problemsbackwards in time to approximately determine the value of being in a particular stage at aparticular point in time. The process continues by iteratively repeating the steps ofapproximating optimal expansion values and routings until the first period is reached. Once thefirst time period is reached, the optimization problems are solved fonvard in time. Thisinformation is used to refine the surrogate models for the value functions, and the backward and

forward processes are iterated until the solutions converge.Objective Four: I will apply the ADP methodology developed in objective three to the

road, HSR, and air California Corridor supernetwork problem solved previously via two stagestochastic programming in objective two. Lastly, my research project will compare and contrastthe two developed methods'

lprease attach additionar sheets as needed.r

Graduate Research Award program Application: 2010-2011 page 1 of 4

APPLICATION FORM

GRADUATE RESEARCH PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By: Administered By:FederalAviation Administration Airport Cooperative Research programU.S. Department of Transportation Transportation Research Board, NãtionalAcademies

PART IV - REFERENCE LETTER ON APPLICANT

(Please Type)

This section to be cornpleted by Applicant:

Name of Applicant. Noreiqa Quentin carlyle Date: '10 Mav 20.10Last First Middle

Applicant's major field of study Svstems Enqineerinq & Operations ResearchTitle of Proposed Research Project:

Approximate Stochastic Optimization Techniques for Transportation Supernetworks

This section to be completed bv referbnce respondent:

NOTE: This applicant has named you as one of two people who know his/her academic and professionalexperience and ability. Your views will help us evaluate this applicant's qualifications for receiving anaward for conducting research on the above project.

Please complete this form, make another copy, and place each copy in a separate envelope. SeaIboth envelopes, stþn each across úäe sea/, and return BOTH envelopes to the appticant forinclusion with the application to be submitted to the Transpo¡Tation Research Board.

Name of Reference Respondent: Mark McDonald

Title: Assistant Professor of Civil Engineering

Organization: Vanderbilt University

Mailing address: 283 Jacobs Hall, Vanderbilt University, Nashville, TN 3723

Email address: _mark.p.mcdonald@vanderbilt.edu_ Phone: _61 5-343-6386

lnwhatcapacitydoyouknowtheapplicant?-GraduateAdvisorHow long have you known the applicant? 2 years

Graduate Research Award Program Application: 2010-2011 Page 2 of 4

PART lV: Page 2

1. Please evaluate the applicant in the following areas as compared with other individuals ofcomparable training, age and experience.

2. Please comment on the ability of the applicant to carry out the proposed research in a timelymanner.

Major Noreiga has the talent, drive, and motivation to carry out this plan of proposed research in atimely manner. lf awarded this scholarship, I am certain that Major Noreiga will make manyoutstanding research contributions and solve some very difficult problems in multimodaltransportation systems planning.

3. Please add any other comments that you consider to be pertinent to the evaluation of theapplicant and that are not covered adequately by your other answers. Attach additional sheet(s) ifnecessary.

Please see attached letter.

CONFIDENTIALITY: The information contained in this reference shall be available to theapplicant or otherwise publicly disclosed

Signature of Reference Respondent:

ì¡ rfclan¡,lin¡ Aboveaverage

AverageBelow

average

lnsufficientopportunity to

observeJUtÐtát tuil t\

(nowledge of major field X

ìesearch skills X)roblem solving skills X

)reativity X

-eadership X

ffritten communication X

)ral communication X

Graduate Research Award Program Application: 2010-2011 page 1 of 4

APPLICATION FORM

GRADUATE RESEARCH PROGRAM ON PUBLIC.SECTOR AVIATION ISSUES

Sponsored By: Administered By:FederalAviation Administration Airport Cooperative Research programU.S. Department of Transportation Transportation Research Board, NãtionalAcademies

PART IV - REFERENCE LETTER ON APPLICANT

(Please Type)

This section to be completed by Apolicant:

Name of Applicant: Noreioa Quentin carlyle Date: 10 May 2010Last First Middle

Applicant's major field of study Svstems Enqineerinq & Operations ResearchTitle of Proposed Research Project:

Approximate Stochastic Ootimization Techniques for Transportation Suoernetworks

This section to be comoleted by reference respondent:

NOTE: This applicant has named you as one of two people who know his/her academic and professionalexperience and ability. Your views will help us evaluate this applicant's quatifications for receiving anaward for conducting research on the above project.

Please complete this form, make another copy, and place each copy ín a separate envelope. Sealboth envelopes, sign each across the seal, and return BOTH envelopes to the applícant forinclusion with the applicatíon to be submitted to the Transportation Research Board.

Name of Reference Respondent: Sankaran Mahadevan

Title: Professor of Civil and Environmental Enoineerino

Organization: Vanderbilt University

Mailing address: Box 1831-8. Nashville. TN 37235

Email address: sankaran.mahadevan @vanderbilt.edu Phone: 61s-322-3040

ln what capacity do you know the applicant? As instructor in 2 qraduate classes. and as Ph.D. committee

member

How long have you known the applicant? 2 vears

Graduate Research Award program Application; 2010-2011 Page 2 of 4

PART lV: Page 2

1. Please evaluate the applicant in the following areas as compared with other individuals ofcomparable training, age and experience.

lr rt¡ta Aboveaverage

Average Belowaverage

lnsufficientopportunity to

observe

r'u19tqt tvI t\

(nowledge of major field xìesearch skills x)roblem solving skills X

)reativity X

-eadership xüritten communication x

)ral communication x

2. Please comment on the ability of the applicant to carry out the proposed research in a timelymanner.

Quentin Noreiga has completed an outstanding list of graduate classes that have strongly prepared himfor the proposed research. He has a carefully thought out and well organized research'plàn, and isreceiving excellent mentoring from his advisor. Mr. Noreiga is intelligent,hard working and'persistent. I

therefore have no doubt that he will be able to complete the proposeð research in a timely mânner. (Seeattached sheet for more details).

3. Please add any other comments that you consider to be pertinent to the evaluation of theapplicant and that are not covered adequately by your other answers. Attach additional sheet(s) ifnecessary.

(See attached sheet)

CONFIDENTIALITY: The information contained in this reference letter shall not be available to theapplicant or otherwise publicly disclosed except as required by law.

Signature of Reference Respondent:

--S-¿^:L- o^r". t/¡zf yoto

Ciuil and EnuironmentalEngineering

vANDERBTLT \/ School of Engineering

VANDERBILT UNIVERSITY

VU Station B #35r83rz3or Vanderbilt Place

Nashville, Tenne ssee 3 7 z3 5 - 18 3 r

Recommendation Letter for Ouentin NoreigaSankaran Mahadevan

Professor of Civil & Environmental Engineering, Vanderbilt University

I have known Quentin Noreiga as a graduate student at Vanderbilt University since August 200g,when he began his graduate student in our department. Quentin has taken 2 graduaie courses with me --Probabilistic Methods in Engineering Design in Fall 2008, and Advanced Reliability Merhods in Spring2009. He has also been participating in a weekly lunch seminar meeting on reseaich topics retatã¿ tõreliability and risk analysis methods for four semesters now. I am the director of a rnuiti-disciplinarydoctoral program in Reliability and Risk Engineering (initiated by an NSF IGERT grant), and euentin isa member of this program, and his dissertation research is supervised by Prof. Mark McDonald. I amserving on Quentin's Ph.D. Committee as a senior member.

My class in Fall 2008 covered topics such first-order and second-order reliability methods,reliability-based design, Monte Carlo simulation (including efficient sampling such us i-portun""sampling, Latin Hypercube sampling etc.), response surface methods, and system reliability. Thå secondclass on advanced reliability methods in Spring 2009 covered topics such as stochasiic processes,advanced surogate modeling techniques, advanced Monte Carlo techniques, time-dependent ieliability,fatigue and other types of degradation, Bayesian statistics, and reliability-based optimization. thehomework assignments and term paper in this class required strong programming skills.

In addition to the above classes, Quentin has taken several other classes that have given himstrong depth in the areas of optimization, and intelligent transportation systems. His proposed ph.D.research proposal is very well organized, focusing on resource allocation and policy decisions underuncertainty for large and complex transportation networks.

Quentin is sincere, hard working, and highly disciplined and organized. He is strongly engaged inthe class discussions, and works relentlessly until the task is done. Due to his U.S. Army experienóe, hehas natural leadership abilities, and has quickly become an example for other graduate students in ourdepartment. His presentation skills are outstanding, and his communication is always precise and clear.He is also very kind and helpful to his fellow students.

In summary, I very strongly believe that Quentin Noreiga has the potential to be an outstandingleader in the transportation engineering field. I am very happy that we have attracted him to Vanderbilt tópursue his graduate studies. He has chosen an excellent research direction, and is strongly prepared for anoutstanding dissertation and timely completion. I therefore enthusiastically recommend Quentin Noreigafor the ACRP Graduate Research Award, and I believe that the award will be an excellent investmenttowards enhancing the nation's transportation engineering leadership by supporting this outstandingstudent.

-- <l-¿"D*>\-Sankaran MahadevanMay 13,2010

www.cee.vanderbilt.edutel û5.322.2697fax û54223365

Graduate Research Award Program Appllcation: 2010-2011 Page 3 of 4

APPLICATION FORM

GRADUATE RESEARCH AWARD PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By'.Federal Aviatlon AdministrationU.S. Department of Transportation

Administered By:Airport Cooperative Research ProgramTransportation Research Board, National Academies

PART V - FACULTY RESEARCH ADVISOR

To be completed by the applicant:

NOTE: ln order to enrich the educational experience gained from your proposed research project, it isnecessary for you to request a faculty member from your university who is famitiar with your researchproiect to act as a research advisor to you during the course of the project. Ptease provide the fottowinginformation and ask the faculty member to complete the form. You shoutd submit it with your application.The research advisor may also be a reference respondent.

Applicant's Name: Noreiga Quentin Carlvle Date: 10 May 20'10LaSt

Title of Proposed Research Project:

First Middle

Approximate Stochastic Optimization Techniques for Transportation Supernetworks

To be completed by the faculty research advisorNOTE: The research product of research award recipients can be considerably enhanced if a facuttymember at the applicant's university acts as an advisor to the applicant during the conduct of theresearch. Therefore, each applicant is required to designate such an advisor who witl be available tohim/her throughout the course of the research project to provide advice as it progresses. When researchpapers by award winners are published by the Transportation Research Board, the faculty member witl beidentified as the research advisor.

Faculty Research Advisor's Name: Mark McDonaldTitle: Assistant ProfessorDepartmentUniversity: Vanderbilt UniversityMailing Address: 283 Jacobs Hall, Vanderbilt University, Nashville, TN 37235Email: _mark. p. mcdonald@vanderbilt.edu_ Phone:

1. Have you examined the applicant's proposed research plan? Yes _X_ No

Civil Engineering

Yes X No2. Do you consider the applicant's research plan reasonable?lf no, please comment.

Graduate Research Award Program Application: 2010-2011 Page 4 of 4

PART V: Page2

3, Do you believe that this appl¡cant can complete the proposed research within the time frameindicated? Yes X No lf no, please comment.

4. will the applicant receive academic credit for this work? yes _ No _. lf yes, pleaseindicate the nature of this academic credit. [Note: Receiving academic credit in no way countsagainst the applicant.l

Major Noreiga will be using the results of the proposed research in his PhD dissertation

5. Please indicate briefly how you plan to monitor and advise on the work of the applicant on thisproject.

I have regular meetings with Major Noreiga to help him identify the gaps in knowledge his researchmust address, help him determine the approaches that will help close these gaps, and help himunderstand the results of the work he has accomplished. These meetings occur at least weekly and moreoften as needed.

6. I am willingaward. Yes

Signature:

to be the research advisorXNo

if the applicant receives this research

o^," {/tz/zo,o

Ciuil and EnuironmentalEngineering

VANDERBTLT \tr School of Engineering

}l4ay 17,2010

Dear ACRP Scholarship Selection Panel:

I am pleased to give Quentin Noreiga my strongest recommendation for the ACRPscholarship. Quentin came to Vanderbilt to pursue a Masters' degree on leave from theU.S. Army, but upon his return to school he became very enthusiastic about hisresearch, so much so that he accomplished the very rare featof extending his leavefrom the Army by two additional years to do a PhD after only initially embarking uponhis Masters degree as a two year corrse of study. I have known Quentin for two yeaïs,and am delighted to have officially served as his PhD advisor for the last year.

Quentin came to Vanderbilt from the U.S. Army with considerable experience inpractical transportation and logistics problems that have to be solved on the front linesof the war in Afghanistan. His prior experience as an Army Operations Researcher andSystems Analyst (ORSA) has made him a very valuable graduate student atVanderbilt.

Since coming to Vanderbilt, Quentin has taken a very technically challenging courseload in optimization, stochastic methods, transportation engineering, and aerospaceengineering. He impressed me in my graduate course in Engineering DesignOptimization, eaming an A in this challenging course that covers nearly every area ofoptimization methods, including linear programming, nonlinear programming, integerprogramming, genetic algorithms, simulated annealing, dynamic programming,network optimization, multidisciplinary optimization, stochastic programming, andreliability-based design optimization. Quentin's term paper resulted in a paper that waspresented at this year's TRB Annual Meeting in V/ashington, D.C.

Quentin has taken on a very challenging and mathematically demanding researchproblem that will have great impact on planning multimodal transportation systems.The methods Quentin is developing will allow transportation systems planners todesign solutions to the gridlock in our nation's airspace by integrating airport capacityexpansion, better suited afucraft", better logistics, projects in other modes oftransportation such as high speed rail, and pricing mechanisms into a commonoptimization framework.

Quentin is a highly motivated individual I believe will be highly successful incompleting the plan of research he is pursuing. He has brought discipline, a strongwork ethic, sincerity, and professionalism to our department. I believe that Quentinwill be a leader in the field of transportation and logistics in the u.S. Army and

VANDERBILT UNIVERSITY

pMB j5r83r www.cee.vanderbilt.eduz3or Vanderbilt Place tel 615.322.2697Nashville,Tennessee 37235-r83t fax615.3zz3365

boyond. I firmly believe that Quentin is deserving of the ACRP scholarship and thatthe ACRP is making a wise investment by choosing to support this outstandingindividual.

Sincerely,

rtuWftrMMark Philip McDonaldAssistant Professor of Civil EngineeringVanderbilt University

u¡ir¡o srerrs MTLTTARY AGADEMY

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Ðate 05/1713010

1 Problem Decomposition Methods for Joint Airport Network Design, Aircraft Design,2 lnd Aircraft Routing3

4567

8 Quentin Noreiga9 Graduate Student

10 261 Jacobs Hall1.r Vanderbilt University12 Nashville, Tennessee 3723513 quentin.c.noreiga@vanderbilt.edut4 (423) s0s-s96715

16 Mark McDonald17 Assistant professor of Civil Engineering1.8 283 Jacobs Hall19 Vanderbilt University20 Nashville, Tennessee 3723521. mark.p.mcdonald@vanderbilt.edu22 (6t5) 974_277t23

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30 Words: 636731 Tables: 232 Figures:233 Total Words:736734

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Problem Decomposition Methods for Joint Airport Network Design, Aircraft Design,and Aircraft Routing

ABSTRACT

5 The American airspace is seriously congested, and is rapidly approaching a state of gridlock. With6 limited resources avqilable to alleviøte congestion, there are a number of solutions to the problem of7 gridlock, though no single qction will be sfficient to solve the entire problem. Resolving this8 problem may involve expanding airport capacity in some limited sense, including new fleets of9 aircraft, and routing them fficiently. At least some of these decisions must be møde under

10 uncertainty in the future demand. Optimization qppeqrs to be a promising tool to better design the

tt air trønsportation network and the vehicles which operate in it, but dfficult problems must be solvedtZ if network design, øircraft design, and aircraft routing problems are to be solved under uncertainty,13 Problems of this type are multi-støge stochastic mixed integer nonlinear programming problems1,4 (MINLP) for which no scalable solution strateg/ exists. This paper proposes a novel decomposition15 methodfor multistage stochastic MINLP. This methodology has reduced a previously intrqctable16 problem in network design to a series of smaller second-stage subproblems which cqn be solved in17 parallel, and the recourse function is approximated using q radial basis function (RBF) netvvork.

18 Because the RBF network is a universal and computationally inexpensive approximator of the

tg recourse function, the first stage problem is solved using only the first stage decision variables by

20 replacing the recourse costs with the RBF network. This decomposition strategt is illustratedfor a

2t problem ofjoint airline network design, aircraft design, and aircraft routing.22

23

1 I. INTRODUCTION2 I.l Motivation3 The US air transportation system is seriously rapidly approaching a state of gridlock.4 With limited resources available to alleviate congestion, there are a number of solutions to the5 problem of gridlock, though no single action will be sufficient to solve the entire problem.6 Resolving this problem may involve expanding airport capacity in some limited sense, including7 new fleets of aircraft, and routing them eff,rciently. At least some of these decisions must be

8 made under uncertainty in the future demand. Optimization appears to be a promising tool to9 better design the air transportation network and the vehicles which operate in it, but some

10 diffrcult problems must be solved if network design, ahcraft. design, and aircraft routingtt problems are to be solved under uncertainty. The U.S. air transportation system is a complext2 large scale system made up of large systems that impact each other. These problems present

i.3 large challenges with interrelated problems that the operations research community has not yet

14 handled.15 The development of a large-scale system, composed of components that are themselves

i,O complex systems, poses a far greater challenge than the delivery of a single monolithic system to

17 meet a set of originating requirements, Several different aircraft (each an independent system)

18 must be coordiqrated to satisfy a global need. A "best" scenario results from determining how the

l-9 aircraft can be used together to best fit the airline's need. If the airline considers only the aircraftzo in its current fleet, integer design variables represent the numbers of each available aircraft type

2L operated on each route , and existing operations research methods are very well suited to this22 type of problem.23 When a new aircraft fleet is introduced to the system, the best use of aircraft in the

24 network clearly depends upon the properties of the new fleet. But in order to design the new

25 aircraft, it is vitally important to understand its mission requirements. These requirements are

26 determined by solving problems in fleet assignment and network design; but, the solution of27 these problems requires knowledge of the aircraft properties. Once the mission is known,

28 conceptual sizing of the aircxaft is itself a complex optimization problem, and is itself nonlinear

29 in nature. Obviously, network design involves long term planning and requires future demand

30 forecasts, which are often uncertain. The overall decision problem can be thought of as a multi-3t stage decision process. Decisions about the design of the network infrastructure and the aircraft

32 must be made initially, and then, the aircraft are allocated at a later point in time to either

33 optimize the revenue of its ticket sales or to minimize the cost of meeting a set of required

34 demands. The objective would be to minimizethe total cost of meeting the demands over all35 stages, or to maximize profrT.. At this point, the integrated problem is no longer a simple problem

36 in systems design. It is a problem in the design of a family of systems (FoS),

37 Decisions such as these are critical when airlines introduce new origin-destination38 routings and fleets to their infrastructure. More quantitative insights to resource allocation, to

39 include personnel and equipment, can provide the Federal Aviation Administration and other

40 governmental stakeholders a means to streamline their efforts. Aircraft designers and builders

41. can also utilize these methods to create vessels catered to complete specialized tasks or to create

42 vessels capable of supporting a wide range of missions.43

44 L2 Problem Statement

L "Develop a new optimization methodology that can solve joint aircraft design and network2 design problems by using decomposition approaches with optimization and analytical reliability3 methods."4 The objective of this paper is to develop a new optimization methodology that can solve5 joint aircraft design and network design problems using decomposition approaches similar to6 those used for two-stage stochastic programming problems, such as Benders decomposition.7 Combining these methods with optimization and analytical reliability methods allows for the8 generation of reliable and effrcient new system designs in a larger FoS context. The synthesis of9 these systems of systems presents different challenges than the complex, but single, large-scale

10 system (e.9. a single airc;:aft, ground vehicle, etc.) designs addressed by curent engineeringII methods.t2 In this paper, we treat the family of systems problem, and in particular the joint aircraft13 design and network allocation problem, as an assignment problem with variable numbers of1,4 assets. Some of the assets have variable properties, and the demands on the entire network have

i.5 uncertainty described as random variables with known probability distributions. The additional16 complexity of FoS design arises, in part, because each of the independent systems perform tasks

t7 that contribute to meeting the overarching global need. In some cases, different systems could18 perform the same basic tasks, making proper system task assignment more challenging. A new19 system that will operate within the system of systems must be designed with the goals ofzO improving the system of systems' global performance. This is different from the traditional21, practice of working to improve the individual system's performance; therefore, in order to22 effectively design a FoS, three questions must be answered. First, the designer must choose the

23 mix of systems to be present in the FoS. Secondly, the designer of a FoS must choose the

24 properties of yet to be designed systems. Thirdly, the designer must allocate tasks to each of the

25 systems in the FoS. These decisions are clearly not independent, and are fuither complicated

26 because they must be made under uncertainty,27 To address these issues, we propose a novel approach to the joint conceptual design and

28 operation of airline physical infrastructure and aircraft, using a multi-stage, stochastic mixed-29 integer nonlinear programming approach. Applying this methodology to a problem of joint30 aircraft and network design under uncertain demands, we propose a decomposition approach that

31 uses sutrogate modeling of the optimal second stage costs. This decomposition approach allows

32 for the solution of practically sized problems, and also allows for the exploitation of parallel

33 computation. We illustrate the proposed methodology with a numerical case study.

34 The rest of the papet is organized as follows. Section 2 reviews the basic methodologies

35 used in solving this problem. These include airline fleet management using integer programming

36 models, aftc;aft conceptual sizing, multidisciplinary optimization, and reliability-based design

37 optimization methods, including a recently developed methodology for solving MINLP problems

38 with reliability constraints. Section 3 gives a statement of the problem, develops problem

39 formulations for the deterministic problem and its six probabilistic variations, and gives results.

40 Conclusions and suggestions for future work are given in section 4.

41.

42 2. BACKGROUND AND LITERATURE REVIEW43 2.I Aircrqft Capacit.v Design44 In determining the profit function, operating costs must be estimated for the aircraft. We

4s will need to determine certain key aircraft design parameters in order to estimate the operating4G cost. This is done using a simplified aircraft performance modeling approach found in Raymer

1

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(/). We will assume that the direct operating cost is proportional to the fuel cost, and require theaircraft to carry enough fuel to serve a design mission 25 percent longer than the longest route inthe network,

'We will use the Breguet range equation.

Wro-W, - ^åh

wro

where W¡ is the fuel weight, wro is the takeoff weight, rR is the range, C is the specific fuel

consumption rate, I/ is the cruise velocity , andLlD is the lift-to-drag ratio. The lift to drag ratiois calculated from

D gCrn . Wro I

WIS ,S qilewhere V//S is the wing loading (a design variable), the parameter C r, is the zero-lift drag

coeffrcient, assumed to be 0.015, A is the aspect ratio, q:Ion', where p is the air density,

fi. and e is the Oswald efficiency factor, which is approximately 0.8 for civil transport aircraft. V/e1.2 will also require that the aircraft be able to take off from a runway of 8000 feet in length at sea

L3 level, thust4 (roP)ocr.gtw)-(wts)>0 (3)

15 where the parameter ToP is the Takeoff parameter, which is approximately 200 for a twin jet16 engine airqaft"using an 8,000 foot runway, o is the relative air density, which is the ratio of the

17 takeoff air density to sea-level air density, C, is the wing maximum lift coefficient, and TlI4 is18 the thrust-to weight ratio.19 The takeoff and empty weights (wr) are related by the equation

20 wr: awffl (4)

2L where ø is taken to be 1 ,24, and the takeoff weight must be equal to the sum of the empty22 weight, the payload weight (assumed to be 200 pounds per passenger), and the fuel weight, thus

23 Wro:Wn +200N oo"+ll¡ (5)

24 Thus, the aircraft design optimization, given passenger, range, and takeoff distance

25 requirements are stated as

5

6

7

(l)

(2)

10

min l4/,

w.r.t.

s,vYro, A,T,Wf

s.t.

Wro -Wt : rffiwro

R- Rr"quir"d > 0

7Cro.W I

WIS S qxAe

(ToP)oC LQIW) - (WlS) > 0

Wo : ow;:tWro : Wn +200N o* *Wf(wro I s) .i, s w.ro I s < (['l/.ro I s)

^"*A,¡n 3 A1A^o,(Tlwro),, <Tlwro < (Tlwro),*

3 It is not desirable to solve this mathematical programming problem for every trial4 solution in a fleet management integer program. For this reason, we will develop a surrogate

5 model which will predict the aircraft's direct operating cost given the required flight range and

6 distance, assuming an optimally designed aircraft. 'We have solved this model for a design

7 mission of 5000 nautical miles with varying numbers of passengers. For simplicity, we have

8 assumed a maximum aspect ratio of 8, Using the results of the optimization and conducting a9 regression analysis, we have determined the approximate aircraft direct operating cost per

10 passenger mile as a function of the number of passengers to be governed by the proportional

t1. relationship12 DoC æ N;9;:3 R'z:0.903 (7)

13 This relationship is developed on the basis of payload requirements ranging from 100 to

t4 500 passengers. However, in making design decisions, it is usually impossible to know the

15 design requirements a priori without some understanding of the operations of the airline,

16 Therefore our design optimization problem must include a model of airline fleet management.

17 'We also note that the conceptual sizing problem is treated as a purely deterministic optimization18 problem.19

20 2.2 Airline Fleet ManagementzI Given a flight schedule and a set of aircraft., the fleet assignment problem determines

zz which type of aircraft, each having a different capacity, should fly each flight segment to

23 maximize profitability, while complying with a large number of operational constraints. The

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objective of maximizing profitability can be achieved both by inøeasing expected revenue, andby decreasing expected costs. Operational constraints include the availability of maintenance atarrival and departure stations (i.e., airports), gate availability, and aircrafr. noise, among others.Assigning a smaller aircraft than needed on a flight results in spilled (i.e., lost) customers due toinsufficient capacity. Assigning a larger aircraft than needed on a flight results in spoiled (i.e.,unsold) seats and possibly higher operational costs.

Literature on fleet assignment models (FAM) spans nearly 20 years and is documented inthe work of many authors. Beginning with Raymer's (/) model for aircraft performance, we usethe aircraft operationing costs as a function of the fuel cost. Based on this we explore methods tominimize cost by varying the aircraft passenger capacity. Previous works include Gopalan andTalluri (2) who considered airline scheduling planning, and Hane et al.(3) who examined thefleet assignment problem. Barnhart et al. (4) combined fleet assignment and the airline scheduleplanning problem, while Barbarosglu and Arda (5) modeled transportation planning by utilizinga two-stage schocastic programming framework. Meanwhile, Abara (d), focused on applyinginteger linear programming to the fleet assignment problem. These works provided theframework and theoretical motivation for our problem decompostition method for airportnetwork design, aircraft capacity determination, and aircraft routing.

2.3 Stochastic Programming with RecourseThe problem of joint network and aircraft design is clearly a two-stage stochastic

programming problem with integer variables. The problem of stochastic programming can besolved as a deterministic mathematical programming problem, albeit a very large one. Two ofthe most common methods of solving large scale integer linear program are the cutting planemethod and the branch and bound method (f. These methods have been applied successfully tolarge integer programs, yet computational expense for even deterministic integer programs islarge.

In recourse models, it is desired to decompose the problem into smaller pieces that can besolved in parallel and then reassembled to solve the overall problem. Benders Decompositionoffers a method to break down the master problem into computationally feasible sections.Benders decomposition has been shown to solve two-stage stochastic linear programs utilizing a

relatively small number of iterations; however, when the master problem involves numerousinteger variables, Benders method becomes excessively computationally expensive (8). BendersDecomposition begins with the formulation of master problem. After an optimal solution isderived for the sub-problem, solutions are developed for each scenario (9),

Another method for use in two stage integer linear programming with recourse is the L-Shaped method (10). This method begins by first separating the master problem from the sub-problems and solving the master problem. Next, the method requires a feasibility check of thesub-problems and if necessary, the addition of a feasibilþ cut. After checking the optimality ofthe sub-problems, addition of an optimality cut is performed if needed.

The Dual Decomposition Method by Caroe & Schultz (1/) is another stochastic integerprogram formulation with recourse. This methodology is quite efficient and has accommodatedmulti-stage linear stochastic programming problems with mixed integer variables. Norkin andErmoliev (12) provide a stochastic branch and bound method which requires a comparison of thesolution space after multiple solutions are derived within the range of the stochastic variables,Areas within the solution space are further analyzed based on concentrated sections within thesolution space.

1 Sampling methods are also used to decrease the computational effort of integer2 programming with recourse by using a smaller subset of the scenarios. Popular techniques3 include uniform sampling methods such as Monte Carlo simulation. Higle & Sen discuss a4 method for stochastic decomposition which uses a cutting plane algorithm for two-stage5 stochastic linear programs with recourse (/3). Ermoliev provides a detailed look at stochastic6 Quasi-gradient methods (/4). These methods show that sampled approximation methods often7 obtain accurate results for stochastic optimization problems while significantly reducing the8 computational expense,9 If not for the nonlinearities involved in aircraft design, previous methods for integer

10 programming with recourse would be adequate to address the problem of choosing which1,1 infrastructure capacity expansions to build and routing the aircraft to meet the uncertain future1.2 demand. However, when the problem of aircraft design is encountered, the problem becomes a13 stochastic mixed-integer, nonlinear program (MINLP) and there are no currently scalable14 methods to solve problems of this type. For that reason, we propose the decomposition approach15 presented in section 3.

I617 3. PROPOSED METHODOLOGY18 3.1 Master Problem and Sub-problemsL9 Before looking at the means to reduce the computational expense of conducting a20 network optimization problem, we first look at the total problem and formulate the master2I problem. In the first stage, we must choose the infrastructure capacity expansions to determine22 the capacity of the (yet to be designed) aircraft. Once the decisions about the infrastructure and

23 aircraft are made, at the future time, the aircraft are routed to minimize cost or maximize revenue24 or profit. When a problem is divided into sections or steps, whether for computational feasibility25 or more detailed analysis, we refer to these smaller sections as sub-problems or stages. The

26 second stage problems are formulated such that given fixed values of the stage one decision27 variables and a realization of the random event, we wish to minimize the recourse cost (or

28 maximize the recourse profit). The first stage problem would then be to minimize the expected

29 recourse costs as a function ofthe first stage decision variables.30 'We note that the second stage problems are much easier to solve than the entire master

3L problem, which becomes intractable rapidly. In fact, we can solve these problems using a

32 number of combinations of the stage one decision variables and random events at relatively33 lower expense using parallel computation. This allows for the approximation of the expected

34 optimal recourse costs as a function of the first stage decision variables using a surrogate model35 developed on the basis of the solutions to the various stage two sub-problems. These optimal36 solutions become 'training points' from which we develop a model to best estimate the line,37 curve, or surface created by the training points. After a model is derived for the expectation of38 the optimal recourse costs given the first stage decisions, it is then used to determine the optimal39 solution points across the range of the random input variable. We propose that using the model40 results in a large computational savings over direct solution of the master problem.4t42 3.2 Training Point Selection and Second Støge Problem Solving43 We utilize a statistical design of experiments to determine the appropriate values of the

44 first stage decision variables to derive the training points for our sample model. Though this4s could be done with many types of experimental designs, we choose Latin Hypercube Sampling46 (LHS) because we wish to use only a small number of points while still covering the range of the

1 first stage decision variables. LHS is implemented by creating equal partitions of the first stage2 variable and taking samples from every partition (15). As such LHS offers a thorough coverage3 of the design space. At each of these points, Monte Carlo simulation is used to sample the4 demands. A number of second stage sub-problems are then solved to determine the expectation.5 We use a penalty function to ensure feasibility in the second stage problems. When the surrogate6 model is used to optimize the first stage decision variables, the optimizer will not choose7 solutions that make second stage sub-problems infeasible.8

9 3.3 Surrogate Modeling of Optimal Recourse Costs10 After the expectations are calculated for each of the training points, a response surface isy. created using any appropriate surrogate modeling technique, including linear regression,t2 Gaussian process models, neural networks, and radial basis function (RBF) networks. In this13 paper, we choose to use RBF networks to build the surrogate model of the expected optimalt4 recourse costs. Radial basis function (RBF) networks typically have three layers: an input layer,15 a hidden layer with a non-linear RBF activation function and a linear output layer. The output,16 <p(x) of the network is thus

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e@) =\a,plllx - c,ll)

where N is the number of neurons (training points) in the hidden layer, c¡ is the center vector for

neuron l, and a¡ àre the weights of the linear output neuron. In the basic form all inputs are

connected to each hidden neuron. The norm is typically taken to be the Euclidean distance and

the basis function is taken to be Gaussian, i.e.

p(r) = exp(-Br2 ) for some P >0 (e)

In addition to the above unnormalized architecture, RBF networks can be normalized.Inthis case the mapping is

(8)

where

f ntll,-''lll

uql*- r,ll¡'1-

=la,uçllx - c,ll)

n(lx - c,ll

(10)

(i 1)

(r2)

Ipft-",11)is known as a "norm alizedradial ¡aris n n"tiorril

pql. -",1þ = .*nf Pll, - ")l'lThe Gaussian basis functions are local in the sense that

o,rY o,o(ll*-t,ll¡ç(x) =

li- pft - c,ll¡ = oll¡[-a

(1 3)

t2 i.e. changing parameters of one neuron has only a small effect for input values that are far away3 from the center of that neuron. RBF networks are universal approximators on a compact subset

4 of 1R.". This means that a RBF network with enough hidden neurons (training points) can5 approximate any continuous function with arbitrary precision, regardless of its shape. The

6 weights a¡, e¡, and B are determined in a manner that optimizes the fit between Fand the data.

7 We have chosen to use normalized Gaussian basis functions in the numerical illustration in8 Section 4.

9

10 3.4 Model Validation1.1.

1.2 It is important to assess the quality of the surrogate model so that one can have confidence13 that it will be accurate enough for its intended purpose. Statistical hypothesis testing is one

t4 approach to quantitative model validation under uncertainty. Model validation metrics based on

15 both classic and Bayesian statistics have been developed for this purpose. Classical hypothesis

16 testing is a well-developed statistical method for accepting or rejecting a model based on an error

t7 statistic (16-22) Validation metrics based on Bayesian hypothesis testing have been developed in18 (23-2s).19 In Bayesian hypothesis testing, we assign prior probabilities for the null and alternative

20 hypotheses; let these be denoted as P(Ho) and P(H") such that P(Hs) + P(H") : 1 . Here /10 .' model effor <

2t allowable limit, and f1,: model error > allowable limit. When data D is obtained, the probabilities are

22 updated as P(Hsl D) and P(H"lD) using the Bayes theorem. Then a Bayes factor.B (26), is defined as the

23 ratio of likelihoods of observing D under H6 and Ho; i.e., the first term in the square brackets on the right

24 hand side of Eq. (a).

25

28

(14)

26 If B > 1, the data gives more support to Hsthan 1/". Also the confidence in He, based on the data,

27 comes from the posterior null probability P(Hg I D), which can be rearranged from Eq. (14) as

P(H ^)B . Typically, in the absence of prior knowledge, we may assign equal probabilities to

P(H)B +t- P(Ho)

29 each hypothesis and thus P(ÉIp) : P(H"):0.5. In that case, the posterior null probability can be further

30 simplified To Bl(B+l), Thus a B value of 1,0 represents 50% confidence in the null hypothesis being true.

31 A Bayes'factor of 3 is the usually accepted criterion (35) for strong supporl of the model's validity. If the

32 Bayes' factor is less than a threshold of acceptance (say three), it is recommended to acquire more

33 training points to improve the predictive power of the surogate. We emphasize that model evaluation

34 should not be done using the training points,

35 In this application, we can let the null hypothesis be that the observed data has a (normal)

36 probability distribution with the conditional mean given by the model's prediction and the conditional

37 variance given by the variance of the residuals. The alternative hypothesis is that the data has the

38 (normal) distribution with the mean and standard deviation equal to that of the observed data.

39

40 3.5 First Stage Optimization4I Once the approximation function for the optimal second stage costs has been constructed,

42 the first stage optimization is solved to find the optimal infrastructure capacity additions and the

43 optimal design of the aircraft. In solving this problem any appropriate solver for a mixed integer

44 nonlinear programming problem can be used, including a genetic algorithm or a branch and

4s bound algorithm. This solution yields the optimal volume across the origin-destination pairs

P(HolD) _ [ P(Dl H )l P(H o)

-P(rI,lD)-tP@n)P@)

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under any given scenario by solving the appropriate second stage problem. Because the missionrequirements of the aircraft. are known, the aircraft designer can now proceed with a detaileddesign of the aircraft.

3.6 Summar:tTo summarize, the proposed decomposition method can greatly reduce the computational

expense of conducting such family of systems problems as joint airuaft design, networkoptimization, and afucraft routing optimizafion under uncertainty. To recap, the steps of theprocess are:

Figure 1: Problem Formulation & Decomposition

Formulate the complete optimizotion master problem.

Partition the ronge of the rondom inputs variobles into computationally troctøble sub-

problems with the second stage problems dependent upon the first stage decision

variobles. lJse o penalty function to ossure feasibility.Perform Lotin Hypercube Sompling to choose the values of the first stage decision

variobles to serve as training points

Solve the sub-problems and use their output combined with their associate inputvoriable datø as training points for the model approximotion.

Construct o surrogate model using (in this instance) Radial Basis Function Networks (or

ony other appropriate surrogate modeling technique).

Validote the model using on appropriate model validation technique (i.e. classical

hypothesis tests or Bayes Foctor).

Solve the stoge one decision problem.

Utilize the parometers of the stoge one decision to perform detailed optimizotion of the

aircroft as actual mission requirements (for stoge two) are now known.

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a

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Portition MP into

sub-problems

Perfom LHS Sdmpl¡ng(establ¡ sh tro ¡ ning po¡nts)

Construct Surrogdte Model(using troining points)

Solve Stoge Two

Problem Us¡ng Stage

One Results

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We will now demonstrate this decomposition approach using a joint networkinfrastructure design, aircraft sizing, and aircraft routing problem.

4. NUMERICAL ILLUSTRATIONFor the sake of illustration, we propose a network topology consisting of 5 airports with

fixed arcs within the network as shown in Figure 2. Each airport has a unique number of gates.Gates facilitate container departures to the other nodes in the network. Aircraft can move withinthe network along prescribed tours, which are combinations of no more than three cities visitedin strict sequence (i,e. ABCA). The starting and ending node are the same, thus preparing thenetwork for its next iteration on the next day. Initially, airport A has 26 gates, airport B has 12gates, airport C has 29 gafes, airport D has 17 gates, and airport E has 21 gates. There is onlyone size of aircraft to be used with a capacity to be determined at the present.

Demand is given in terms of passengers wanting to move from one origin to a

destination. Future demand is uncertain at the þresent, but known at the time the aircraft are to be

routed. As is the case in the United States air transportation system, the network is congested,and new infrastructure is required to alleviate congestion in the network. In order toaccommodate an increase in passenger demand, we add additional gates to the network tomaintain an acceptable ratio of departures to gates. Note that the issue is not how the capacity isexpanded; but, simply the need for capacity expansion in the use of mixed integer programming.For our network, the departure to gate ratio must not exceed five aircraft per gate per day. Totalfuture demand for each origin-destination pair is assumed to be normally distributed with themeans and percentages of total demand given in Table 1.

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Figure 2: Test Network

1.2

OD Pa¡r Demand % of Total Demand

AB 3176 4.66%AC 3A47 5.64%AD 3847 5.640/.

AE 5130 7,52%BA 2992 439%BC 3654 5.3716

BD 2626 3.85%BE 2504 3.67%

CA 3298 4,8?%

CB 3481 5.t0%CD 4153 6.O9%

CE 3481 5.t0%DA 3664 5.37%

DB 232L 3.40%DC 3t76 4.66%

DE 3970 5.82%

EA 2809 4.12%

EB 2626 3,85%

EC 4153 6.09%ED 3298 4.81%

Total 68215 L00%I2

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Table 1: Demand Distributions of Each OD Pair

In our example problem, demand is served by direct flights only. All aircraft areassumed to travel at I00Yo capacity. Network cost is derived from the number of aircraft movingalong each tour, where each tour has a given cost based on the length of the tour and the capacityof the aircraft, according to the formula using the result derived from section2,T:

Tour Cost : k*Length*No*-ot3 *Noo, (1s)

Airport capacity is determined on the basis of the number of aircraft departing each nodeand the number of gates at each node. The value used to describe network capacity is the ratio ofaircraft per gate. This network does not take time into consideration. This network is assumedto take place in one period of time, here assumed to be one day.

There are 80 tours in this network. An arc in the network is serviced by a tour if and onlyif the arc is in a cyclic permutation of the airports in the tour. For instance, tour ABC servicesarcs AB, BC, and CA, but not AC, CB, or BA. Hence the incidence relationships are summarizedby Kroneker delta functions of the form ô,Ü, where t is the tour and i andj refeito the starting andending airports, respectively, of the arc. It is assumed thatat most six aircraft can fly the sametour on any given day. The master problem can therefore be formulated as

22

23

24

min z = cr^u(Go + Gb + G, + Go + G,) + EIR(N,,N po,.Go,...,G")* r(¡r, ,N oo,,Go,...G"J)

with respect toN,,N po",Go,..,,G"

subject to:

(1 6)

25

26

27

28

.õiN,36

Sr=I,.rN,*Npo,

13

>D¡jVieI,ieJ (r7)

T

2

1

100< Np"", < 5oo

r< Go....,G, < 10

Z c,,cu,G",Gd,G, < 25

Nt, Go,..., Gu integerAll decision variables nonnegative

wherecgø": cost of gate addition,Go,...G, : Gates added at airport A,..,8R:Recourse Costs from Stage 2 Problem (Operating Costs Only)N¡: Number of times a tour is flown per dayS¡ : Supply (in pax) þr travel from airport i to airport jD¡¡ : Demand (in pax) for travel from airport i to øirport jP:Penaltyfunctionfor exceeding ollowable departure to gate ratio, calculated as

p * Max (0, Departur es/Gates -5 )

V/hen we attempted to solve the full master problem with only one scenario (where thedemand takes on its mean value) we found that the nonlinear nature of the cost of the routes as a

function of the cost made this problem computationally intractable. The computationalcomplexity of this problem would only grow as the size and complexity of the network grows.

This clearly shows the necessity of the proposed problem decomposition.We begin our description of the proposed decomposition with the statement of the stage-

two sub-problem. In the stage two sub-problems, we wish to minimize the operational cost of the

airline and the cost of any penalties incurred as a result of exceeding the allowable departure togate ratio. Thus given values for the stage one decision variables Go, ...,G, and Nuo, and a

realization of the demand, we wish to solve the following optimization problem:

subject to

min*R(1/,)+P(/V,)

0<N,<6 YieI,jeJ

su :I,.rN,*No**6'l >DùYieI,i eJ vieI,i eJN¡ integer

(1 e)

(20)

All decision variables nonne gative

We solved this model for 25 different stage one training points with multiple demand

scenarios selected randomly for each training point. The average computational expense ofsolving the stage two problem was 80 seconds. Upon collection of the training dala, a radial basis

function network with Gaussian basis functions was constructed. To perform model validation,the model was estimated using only 17 of the training points (chosen by random selection), and

the other I points were used as validation points. Bayesian hypothesis testing yielded a Bayes

Factor of 3.07, signifying that the model is acceptable for the desired purpose.

(18)

5

6

7

8

9

10

1,1

12

1,3

1,4

1.5

16

t718

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

353637

38

39

4041

42

43

t4

T

2

3

4

5

6

7

8

9

10

tt12

L3

T4

15

16

t718

19

20

1'23

2425

26

27

28

One important and intuitive result of solving the stage two sub-problems was hnding thatthe number of capacity expansions required was highly dependent on the size of the aircraft usedin the network. In general, while the larger afucraft tend to be more expensive to fly, the cost perpassenger mile is significantly less, and the required number of infrastructure additions was less.

After solving the stage two sub-problems and building and validating the surrogatemodels, we move on to solving the stage one problem. In the stage one problem, we minimizethe total cost of adding infrastructure to the network, where infrastructure additions are definedas the number of gates at the nodes, plus the expectation of the optimal recourse and penaltycosts. Five design variables mark the number of additional gates required to support the networkdemand, and are integer variables, and the sixth variable, the capacity of the aircraft, is

continuous. The constraints bound the capacity of the aireraft and the number of gate additions.All variables are assumed to be non-negative. The stage one sub-problem is

minz =Cro,"(Go+Gb+G"+Go +G")+ ¿tR(N,,Noor,Go,...,G")+ P(N,,N0,,,Go,...G"y¡)

with respect to

subject to:

N por,Go,"',G,

1oo<Np",, <5oo

l<G,,...,G, <70

I G".Gu,G,,Gd,G" < 25

(21)

(22)

(23)

(24)

29

30

31

32

33

34

No,..., G" integerAll de ci s ion v ariqb I e s nonne gativ e

The symbol ãdenotes that the recourse costs are calculated using the surrogate model. The

results of the stage one optimization problem are shown in Table 2. The computational cost ofsolving this stage one sub-problem was 0.51 seconds

Variable OotimalValue

Gu 1

G5 10

G.

G¿ l0G"

Nn,, 150

Table 2 Optimal Stage One Decision Variable Values

5. CONCLUSIONSThis paper has provided a new decomposition strategy for solving stochastic mixed-

integer nonlinear programming problems with recourse. These types of problems are

encountered in the design and analysis of Family of Systems problems. Such FoS problems are

15

L encountered in the joint optimization of aircraft, airport network design, and airline routing.2 These three disciplines have been traditionally treated separately, but they are clearly3 interdependent, and significant improvements to the performance of the total air transportation4 system can be realized if these disciplines can be integrated. This paper has illustrated the

5 potential of this decomposition strategy to leverage parallel computation and render

6 computationally tractable a class of problems which heretofore has been computationally7 intractable.8 This decomposition has been based on the creative use of surrogate models with unique9 properties, in particular radial basis function networks. As RBF networks are universal

10 approximators on a compact subset of lRt, an RBF network built on the basis of enough training1.1. points can approximate any continuous function with arbitrary precision at any point in itst2 domain, regardless of its shape. This gives an RBF tremendous potential in modeling functionsi-3 with uncertain shapes, such as the expected recourse function, which have no closed form and

L4 are computationally expensive to evaluate. While the expected recourse function may not15 necessarily be continuous, RBF networks can help identify regions of the design space that

16 warrant further exploration in greater detail. However, when surrogate models are introduced, it!7 is necessary to validate them to assure that they achieve a level of accuracy commensurate withi"8 the desired use. This paper has illustrated the use of Bayesian hypothesis testing to validate the

tg RBF network used to evaluate the recourse function. The RBF network is extremely inexpensive

zo to evaluate, and it enables an efficient solution of the first stage problem while still considering

2t recourse.zZ The ability to approximately solve such complicated problems in joint network and

23 vehicle design gives airlines the ability to make better informed decisions regarding where to

24 invest in additional infrastructure, how to make better decisions regarding which aircraft to

2s introduce into its fleet, and how to manage its personnel and equipment under uncertainty, It may

26 also gives the FAA and other governmental decision makers insight on where to invest in27 additional infrastructure, and also can give insight on how best to incentivize airlines to purchase

zg aircraft that will be best for the overall system. It also gives aircraft designers and builders the

29 ability to produce more profitable aircraft for their own businesses and the airlines.

30

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