MULTI-OBJECTIVE AIRCRAFT OPTIMIZATION FOR MINIMUM …aero.stanford.edu/Reports/multiobjectiveaircraftandroute... · · 2008-10-01MULTI-OBJECTIVE AIRCRAFT OPTIMIZATION FOR MINIMUM

26th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

MULTI-OBJECTIVE AIRCRAFT OPTIMIZATION FOR MINIMUMCOST AND EMISSIONS OVER SPECIFIC ROUTE NETWORKS

Geoffrey C. Bower, Ilan M. KrooStanford University, Department of Aeronautics and Astronautics

Keywords: Multi-Objective Optimization, Aircraft Design, Environmental Impact

Abstract

Historically, maximizing profits for corporateshareholders has been the primary goal for air-craft designers. Due to climate change concerns,environmental performance is quickly becominga major design focus. A methodology to designone or more aircraft to satisfy demand on a givenroute network is described. The objectives are tominimize direct operating costs, CO2 emissionsand NOX emissions. A hierarchical decomposi-tion is used with discipline-specific optimizationalgorithms. A modified version of the NSGA-IImulti-objective genetic algorithm is implementedin the system level aircraft design subspace. TheCPLEX integer solver is used in the fleet assign-ment subspace. Results are presented for a testproblem that involves designing a single aislecommercial aircraft for a route network consist-ing of 4 cities and 8 route segments. A Paretofront is found and the characteristics of the op-timal aircraft designs and fleet assignment solu-tions are examined. The results show a definitivetrades between operating costs and NOX emis-sions and CO2 emissions and NOX emissions.Operating costs and CO2 emissions are stronglyrelated, with little if any trade off evident. All de-signs show benefits from using higher bypass ra-tio engines and thicker, higher aspect ratio wingsthan today’s single aisle commercial aircraft.

1 Introduction

The environmental impact of commercial avia-tion is quickly gaining importance in the aircraft

design community as well as with policy mak-ers around the world [1, 2, 3]. As of 1992, avia-tion accounted for roughly 2% of the total globalanthropogenic carbon emissions and 3.5% of thetotal anthropogenic radiative forcing. As air traf-fic is expected to grow at approximately 5% peryear for the foreseeable future, these fractions areprojected to increase to about 3% and 5%, respec-tively, of the global totals by 2050 [4]. Quantify-ing the environmental impact of an aircraft de-sign is difficult due to difficulties in modeling thetemporal and spatial distribution of emissions,the complex chemical interactions, and uncer-tainties in global atmospheric simulations [5, 6].

The primary constituents that have an influ-ence on climate change are CO2, NOX , H2O,SOX and soot [7]. NOX affects climate throughfeedbacks to O3 and CH4 concentrations. Waterand particulate emissions influence the formationof persistent contrails and cirrus clouds. Sulfateemissions have a relatively minor impact on cli-mate change compared to the other exhaust con-stituents, but contribute to acid rain. Noise gener-ated during takeoff and landing is also a concernto aircraft designers and those living and workingnear airports.

The true external costs of various pollutantsand noise are not currently well understood, butsome attempts have been made at quantifying itfor specific locales [2]. As such, there is no sim-ple global objective function weighting the costsof CO2 emissions, NOX emissions, contrail for-mation, noise and other emissions. Past studieshave shown that improving environmental perfor-mance of aircraft often results in higher operating

1

GEOFFREY C. BOWER, ILAN M. KROO

costs and/or reduced performance as measured intraditional parameters such as cruise speed. Forthis reason, aircraft design for reduced environ-mental impact lends itself towards finding an op-timal set of solutions [8]. A Pareto front of op-timal solutions can be generated using a multi-objective optimization algorithm that allows sys-tem level trades between the various objectives tobe analyzed before a final design is selected.

Ideally, aircraft designers choose an objectivefunction directly related to profit potential for theaircraft manufacturer such as return on invest-ment (ROI) or net present value (NPV) [9]. His-torically, surrogates which are easier to computehave been used, such as direct operating costs andmaximum takeoff weight [10]. The value of theselected objective is then minimized for a singledesign mission (a specific payload and range) ora weighted combination of a small set of designmissions.

When assessing the environmental impact ofaviation, the air transport system as a wholeneeds to be considered since overall emissionsare what impact climate change. Factors such asaircraft utilization and routing become very im-portant in terms of total emissions and operatingcosts. This is a change from the historical air-craft design philosophy which focuses on mini-mizing costs on a small set of missions. Therehas been some recent work on the design of air-craft for single hub route networks, but it did notincorporate environmental performance as an ob-jective or constraint [11].

An approach is presented in this paper that al-lows the aircraft designer to optimize a set of oneor more aircraft at a conceptual design level for aspecific route network. The objectives of interestinclude the economic and environmental perfor-mance of a fleet over the route network. A de-scription of the problem and some solution tech-niques are described in section 2. A small scaletest problem is presented in Section 3 and resultsare shown in Section 4. Conclusions and futurework are discussed in Section 5.

2 Problem Formulation

The primary purpose of the commercial avia-tion industry today is to move people and goodsaround the world both quickly and economically.An aircraft manufacturer is in business to makemoney for its share holders by creating prod-ucts and services to meet the demand of airlines.Governments around the world are also stakeholders in the industry to ensure public safetythrough regulations and to manage the flow oftraffic through their airspace. In the past, someof these government regulations have taken theform of limits on noise and emissions around air-ports [12, 13]. With expanded interest in anthro-pogenic climate change today, there is discussionof expanding emissions regulations to encompassthe complete flight regime or to enforce an eco-nomic penalty for emissions [14]. The true exter-nal cost of these emissions is not well understoodat this time; however, it is thought to be signifi-cant [2]. Therefore, it is fortuitous for aircraft de-signers to consider emissions performance dur-ing the conceptual design process to understandand convey to policy makers the tradeoffs be-tween economic performance and environmentalperformance.

There are many choices to be made by theaircraft designer to meet market demand. Theprimary choices are the size and performance ofthe aircraft themselves. The route network overwhich the aircraft operate also has a large impacton the economic performance and total emissionsfor given market demand, but this is largely dic-tated to the aircraft designer through airline de-mand. In this study, we choose to consider boththe route network and aircraft design simultane-ously. By considering both facets it is thoughtthat significant gains can be made in environmen-tal performance of the fleet with a small effect oneconomics. To make the problem tractable, wewill assume that there is a known demand for allof the city pair segments in a specific route net-work. This problem is posed as a multi- objec-tive optimization problem to design one or moreaircraft types to satisfy passenger demand whileminimizing both operating costs and emissions.

2

Multi-Objective Aircraft Optimization for Minimum Cost and Emissions over Specific Route Networks

2.1 Objective Functions

This section describes the choice of objectivefunctions considered during the multi-objectiveoptimization for both economic and environmen-tal performance.

2.1.1 Operating Costs

The first objective is related to the economicperformance of the aircraft. There have beenmany economic performance metrics and surro-gates used in the past: maximum takeoff weight,direct operating costs, total operating costs, ROI,and NPV to name a few. For this problem, we areconsidering the performance over an airline routenetwork, so using an objective which directlyrelates to airline costs is logical. We choosethe direct operating cost plus interest (DOC + I)method described in Ref. [10]. It is both easy tocalculate and adequately reflects the cost to oper-ate an aircraft for an airline on various routes.

The DOC + I method breaks down the op-erating costs into ten components: Flight Crew,Cabin Crew, Landing Fees, Navigation Fees, Air-frame Maintenance, Engine Maintenance, Fuel,Aircraft and Spares Depreciation, Insurance, andInterest. To enable the calculation of costs overthe complete route network, these costs were fur-ther decomposed into their dependence on blocktime, flight time, flight cycles, fuel used and num-ber of aircraft. The cost quantities related tothe aircraft are the cost per block hour ($/bhr),cost per flight hour ($/ f hr), cost per flight cy-cle ($/ f light), cost per year ($/yr) and fuel cost($/lb). To calculate the total DOC + I the fol-lowing quantities are also required relating tothe operation of the aircraft over the route net-work: total flight hours ( f hr), total block hours(bhr), number of flight cycles ( f lights), numberof aircraft (aircraft) and total fuel burn ( f uel) forthe network. The final formula for calculatingDOC + I is given in Eq. (1).

2.1.2 Emissions

As discussed previously we are interested in air-craft emissions that affect climate change. The

most influential emissions are CO2, NOX andcontrail formation. For this study we are onlyconsidering CO2 and NOX emissions as objec-tives since they have a significant impact on cli-mate change and are simple to model. CO2 emis-sions are directly proportional to the total fuelburn. NOX emissions are related to the designof the combustor, the fuel burn, and details of theengine cycle, primarily the overall pressure ratio.Other pollutants for which there are adequate pre-dictive models could be included as objectives aswell. Modeling contrail formation is also pos-sible, but requires detailed knowledge of localatmospheric conditions along the various flightsegments [15]. This type of modeling is beyondthe scope of the current study.

Multiple event noise metrics estimate the im-pact of a series of noise events. An accurate noisefootprint along with the land use around airportsis required to measure the actual effect of noiseon the population around a given airport. Thistype of modeling is also beyond the scope of thisconceptual level design study. It is recognizedthat noise is an important metric for aircraft de-sign today and often drives design decisions [16].For this reason, noise will be treated as a con-straint in this study by requiring the aircraft tosatisfy the ICAO stage 4 noise regulations.

2.2 All-at-Once Formulation

Details of the various analyses and optimizationalgorithms will be described in subsequent sec-tions. Definitions of some important terms fol-low. A segment refers to a flight path in the routenetwork connecting two airports. A route refersto a series of segments taken by a passenger fromtheir origin to destination or a series of segmentsflown by a single aircraft during a day. When re-ferring to "routes" we will mean aircraft routesfrom now on unless otherwise noted. Parametersand variables with subscript i refer to segmentsand those with subscript j refer to routes. Sub-script k refers to the aircraft type if there is morethan one being designed.

An all-at-once approach to this multi-objective optimization problem is outlined in Eq.

3


(2-4).The objectives are DOC + I for a day, to-

tal CO2 emissions for a day and total NOXemissions for a day. The design variables in-clude aircraft and engine parameters for each air-craft type, operating conditions for each aircrafttype, the number of aircraft of each type fly-ing each route and the average number of pas-sengers on each aircraft type for each segment.Here Aircra f tVariables, EngineVariables andOperatingConditions refer to sets of variablesthat can be used to specify the aircraft, engine,and operating conditions, respectively.

Constraints sets include aircraft design con-straints and fleet assignment constraints. Typ-ical aircraft design constraints include takeofffield length and 2nd segment climb gradient,among others. The fleet assignment constraintsinclude constraints related to route compatibility,demand, airport capacity and continuity of air-craft from day to day. Route compatibility de-termines if a given design can fly the specifiedroute. Continuity of aircraft from day to day re-quires that the same number of airplanes of eachtype start/end the day at each airport so that theschedule can be repeated.

The required analyses include calculating thevarious cost components, fuel burn, and NOXemissions for each aircraft type on each seg-ment with the average passenger load specifiedby paxi,k. Fleet assignment parameters also needto be computed along with all of the constraintvalues.

Immediately, this formulation should appearintractable. It is a constrained, multi-objective,mixed-integer, non-linear global optimizationproblem. The only technique that might be ableto solve the problem as formulated is a multi-objective genetic algorithm (MOGA). However,the large number of integer fleet assignment vari-ables and constraints indicate that finding a fea-sible (let alone optimal) solution to the completeproblem will be difficult. Another formulationshould be sought to take advantage of differentoptimization algorithms to more efficiently solvedifferent parts of the problem.

2.3 Hierarchical Formulation

Integer programming methods are well suited tosolving the fleet assignment portion of the prob-lem. MOGAs are one of the best classes ofmethods for solving multi-objective, non-linearglobal optimization problems. Multi-objectivealgorithms which generate a single objective byweighting all of the objectives can also be used.By decomposing the problem into subspaces witheach having its own optimization algorithm andanalyses, the total computational time can be re-duced compared to the all-at-once formulation.However, compatibility between the shared vari-ables and targets in the various subspaces mustbe maintained, increasing the problem size. Inaddition, for each design at analyzed at the sys-tem level a complete subspace optimization mustbe performed.

2.3.1 Initial Formulation

A hierarchical optimization methodology was de-rived to take advantage of our knowledge of theproblem. The optimization problem formulationfor the system level aircraft design subspace ispresented in Eq. (5-7). The fleet assignment sub-space optimization problem follows in Eq. (8).

The system level aircraft design subspacepasses shared variables and targets to thefleet assignment subspace. These includeOperatingConditionsk, f lightsi,k, aircra f tk,paxi,k, bhrk, f hrk and one aircraft designvariable, the capacityk. The fleet assignmentsubspace returns the value of its objectivefunction, J f leet . The primed values in the fleetassignment subspace the are the fleet assignmentsubspace instances of the shared variables orvalues computed in the subspace.

There are still a number of issues with thisformulation. Requiring the average number ofpassengers on each segment as a shared variablepotentially doubles the system level aircraft de-sign subspace variables in the limit of a largeroute network. It also makes the integer program-ming problem non-linear, which is beyond thecapability of the available solver as will be dis-cussed later. The inclusion of the aircraft capac-

4


ity and operating conditions in the fleet assign-ment subspace also make the integer program-ming problem non-linear. In the system level air-craft design subspace the analyses include cal-culating the various cost components, fuel burnand NOX emissions for each aircraft type on eachsegment. Aircraft design related constraints arealso analyzed in this subspace. The fleet assign-ment subspace tries to match target values for thenumber of flights on each segment, total flighthours, and total block hours as well as the targetvalues for the shared variables mentioned previ-ously. The fleet assignment local variables are thenumber of aircraft flying each route and the localcopies of the shared variables.

2.3.2 Simplified Formulation

In order to use a linear integer programming al-gorithm all of the non-linearities in the subspaceobjectives and constraints must be eliminated.The operating conditions impact which routes arecompatible with a given aircraft design and thecapacity makes the passenger demand constraintnon-linear. In addition, we would like to elimi-nate the variables indicating the average numberof passengers on each flight segment in order tosimplify the system level aircraft design subspaceoptimization problem.

The subspace degrees of freedom from theshared variables are only necessary if the sub-space problem is infeasible. By fixing the op-erating conditions and capacity in the fleet as-signment subspace we can make the objectiveand constraints linear. We are able to fix thesevariables in the subspace because we can en-sure fleet assignment feasibility by merely addingmore aircraft.

In order to do this some assumptions mustbe made. If the fuel burn, emissions and airportcompatibility are calculated at maximum payloadthen the paxi variables can be eliminated. Thisassumption is equivalent to assuming a load fac-tor of one. It should be a reasonable assump-tion since the optimal fleet assignment solutionsshould tend towards high load factors to mini-mize both operating costs and emissions. How-

ever, the calculated total costs and emissions willbe slightly pessimistic. Post optimality analysiscan establish the sensitivity of this assumptionand estimate a correction based on the actual loadfactors. For this study no correction was madeand the pessimistic solutions were used.

The system level aircraft design subspace op-timization problem incorporating these changesis shown in Eq. (9-11). The fleet assignmentsubspace optimization problem written in its sim-plest form is given by Eq. (12).

The system level aircraft design subspaceonly passes targets for the number of flights ofeach airplane type on each segment and the to-tal number of aircraft of each type to the fleetassignment subspace. The fleet assignment sub-space returns the value of its objective function,J f leet . The primed values in the fleet assignmentsubspace the are values computed in the sub-space that try to match the targets. The f hr andbhr compatibilities in the objective function wereeliminated since they only depend on f lightsi,kfor a given capacity and cruise speed, both ofwhich are now fixed in the fleet assignment sub-space.

2.4 System Level Aircraft Design Subspace

This section details the system level aircraft de-sign subspace analysis methods. A preliminarydesign tool, PASS (Program for Aircraft Synthe-sis Studies), is used for all aircraft design anal-yses [17]. Given a set of aircraft design vari-ables PASS calculates aircraft performance overa specified mission profile. The modularity al-lows for models of differing fidelity to be incor-porated depending on the application. A newpropulsion model was added for this study toadd degrees of freedom to the engine design tomore accurately reflect the impact of engine de-sign variables on emissions.

The engine model is based on the twin spoolturbofan analysis methods described in AppendixG of Ref. [18]. This method includes the ef-fects of turbine blade cooling, power extraction,and bleed air. The engine is analyzed on-designat the cruise conditions to obtain specific perfor-

5


mance. It is then analyzed off-design at sea levelstatic conditions and sized to meet takeoff thrustrequirements. Thereafter, the thrust and specificfuel consumption can be determined at any oper-ating condition. An improved weight model forthe engine is also included [19].

The engine carbon dioxide emissions are di-rectly related to fuel burn. For each kilogram ofjet fuel that is consumed, 3.13 kg of CO2 are re-leased into the atmosphere. An empirical modelfor the NOX emission index based on curve fitsof existing engine data was used. The primarydrivers of NOX emissions are the details of thecombustor design and the temperature and pres-sure at the burner entrance. The temperature andpressure are primarily a function of the overallpressure ratio (OPR) [3]. Simple curve fits ofemission index (EINOX ) vs. OPR were estab-lished from data in the ICAO emissions data-bank for modern turbofan engines [20]. The en-gines included in the fits are the CFM56-5Bx/2P,CFM56-7B2x/2, CF34-8C, GE90-110B, GE90-115B, the Rolls Royce Trent series, PW4164,PW4168A, PW6122A, and PW6124A. All ofthese engines use some form of a low emissionscombustor. While future engines may have loweremissions indices than today’s engines, the trendswith overall pressure ratio should be consistent.Figure 1 shows the fits of EINOX vs. OPRfor the four certification points. The fit is de-cent for takeoff and climb conditions, but poorfor approach and idle. Fortunately, the approachand idle conditions have the smallest indices andonly represent a small portion of the total flighttime. Emissions indices are not available forcruise conditions since cruise is not a certifica-tion point. The cruise emission index is taken tobe (0.45)0.7 = 0.73 of the sea level static valueof the climb emission index based on recommen-dations in Ref. [21]. In the future, an improvedcruise NOX model should be incorporated.

2.5 Fleet Assignment Subspace

The fleet assignment subspace attempts to matchthe target values of the parameters passed fromthe system level aircraft design subspace while

satisfying local constraints. These parametersare the total number of flights on each segmentand the total number of aircraft required to flythe network. The variables in the fleet assign-ment subspace are the number of aircraft flyingeach possible route. A system level aircraft de-sign subspace analysis routine determines whichroutes are compatible with the specific aircraftdesign being passed to the fleet assignment sub-space based on the calculated cruise speed of thedesign.

This method for the fleet assignment does notdeal with the explicit scheduling of each aircraftthroughout a day as this would dramatically in-crease the size and complexity of the problem;it only determines what aircraft routes should beflown during the day. There is a minimum timerequired to complete each route, and any excesstime can be used to vary the start time, end timeand/or layover times.

As an example, assume there are 12 useablehours in a day, from 6 am to 6 pm. An aircraftis to fly a route from city A to city B and thenback to city A during a day with each flight last-ing 4 hours. If one hour is required for a layover,then the aircraft requires a minimum of 9 hoursto complete its route. The first flight could leaveat 6 am and the second at 11 am ending the dayat 3 pm, or the first flight could leave at 9 am andthe second at 2 pm ending the day at 6 pm, orthe first flight could leave at 6 am and the sec-ond flight at 2 pm ending the day at 6 pm. Thelast scenario has an extended 4 hour layover inthe middle of the day whereas the first two havea 3 hr buffer at the beginning or end of the day.Therefore there is some flexibility in the actualscheduling. Any scenario which has the sum ofthe buffer times at the beginning and end of theday and any additional layover time totaling lessthan 3 hours would be feasible. The describedmethod captures the important information forcalculating operating costs and emissions with-out explicitly scheduling the flights.

6


2.5.1 AMPL

The fleet assignment optimization problem wasset up using AMPL (A Modeling Language forMathematical Programming) [22]. The AMPLmodeling language is convenient for formulat-ing problems of variable size that are based onsets and indexing. It is primarily used to setuplinear, integer and quadratic programming prob-lems. There are a number of available solversthat are tailored to various problems types. First,a model file is defined including names for allof the sets, variables, parameters, objectives andconstraints. Then to solve a specific instance of aproblem, a data file containing values for the setsand parameters is generated and a solver is spec-ified. The output values of interest can be sent toa text file.

2.5.2 Calculations

The fleet assignment subspace has one simpleanalysis associated with it. Before the optimiza-tion begins, a set of all the feasible routes is gen-erated. The fastest cruise speed achievable duringthe optimization (corresponding to the maximumcruise Mach number and the minimum cruise al-titude) is calculated. Then, based on the availabletime in the day for operations and an average lay-over time between flights it is possible to calcu-late whether or not a given route can be flown.Once the set of all possible routes is determined,the minimum cruise speed required to completeeach route is calculated and saved in an array.Other parameters that are derived from the set ofpossible routes are the start city, end city, and thenumber of times each segment is flown on eachroute. These parameters are saved and added tothe data file.

During the optimization run, each aircraft de-sign can be checked for compatibility with eachroute. If the cruise speed is greater than the mini-mum cruise speed required, then the route is fea-sible. Other feasibility checks can be added whenmultiple aircraft are being designed, such as take-off field length and maximum range.

2.5.3 Variables, Objectives, and Constraints

The local variables for the fleet assignment prob-lem are the number of aircraft of each type flyingeach route, f lights j,k. The objective function forthe fleet assignment subspace is to minimize theabsolute value of the difference between the tar-get values and local values for the total number offlights on each segment and the total number ofaircraft. Slack variables are required to linearizethe absolute values in the objective function.

The first set of constraints requires the avail-able capacity to exceed the demand on each seg-ment. Aircraft continuity for each day is alsoimposed. A further basing constraint can be im-posed by specifying a maximum number of air-craft of each type allowed to start/end the day ateach airport. The final constraint that can be im-posed places an upper bound on the maximumnumber of aircraft that are allowed to fly a givenroute during a day. The final formulation of thefleet assignment subspace problem is given byEq. (13-18).

2.6 Optimization Methods

This section describes the optimization algo-rithms used for each subspace. The system levelaircraft design subspace contains multiple objec-tives, is non-linear, non-smooth and may containlocal minima. The multi-objective algorithmsconsidered for this included weighted sum meth-ods and MOGAs. The weighted sum methodstypically utilize gradient based methods and re-quire multiple optimizations to be run to gen-erate a Pareto front. The algorithm choice wasnarrowed to a MOGA for this study since ge-netic algorithms (GAs) typically perform bet-ter on global optimization problems and becausethey solve for the Pareto front in a single opti-mization run. A genetic algorithm was also se-lected since it is able to better handle disconti-nuities in the design space and integer variables[23]. The CPLEX integer programming methodwas selected for efficiently solving the fleet as-signment subspace problem [24].

7


2.6.1 Multi-Objective Genetic Algorithms

There have been a large number of MOGAs de-veloped over the years, starting with the VectorEvaluated Genetic Algorithm (VEGA) in 1984.Two of the more recent algorithms that were con-sidered for this study were the Non-DominatedSorting Genetic Algorithm II (NSGA-II) andthe Strength Pareto Evolutionary Algorithm 2(SPEA2) [25, 26]. Both of these algorithms areelitist and improve upon the previous generationof MOGAs by eliminating explicit niching pa-rameters to achieve an even spread of solutionsover the final Pareto front.

Ultimately, NSGA-II was selected since it re-quired fewer function evaluations on a numberof constrained test problems, at the expense ofslightly worse performance in evenly spreadingdesigns over the final Pareto front. This algo-rithm is also easier to modify for handling con-straints and controlled elitism based on sugges-tions in Ref. [23]. The algorithm selects so-lutions through a domination sorting procedure.A solution dominates another solution if the firstone is no worse than the second in all objectivesand better than the second in at least one objec-tive. The set of non-dominated solutions is thePareto front of a given population. These solu-tions are assigned rank 1. Ignoring this set of so-lutions, the next non-dominated set is determinedand given rank 2. This procedure is repeated untilall solutions are ranked. The fitness assignmentvalue of a design is equal to its rank.

NSGA-II uses binary tournament selection toselect the parent population for the next gen-eration. In each tournament, the solution withthe lower rank is selected. If two designs areof the same rank, then the solution with fewernearby solutions is selected. This algorithm usesa crowding distance estimate in the objectivespace that is equal the average side length of thecuboid formed by its nearest neighbors in eachobjective. Two point crossover and bitwise mu-tation are then used to generate the child popula-tion from the parent population. The child pop-ulation is then combined with the parent popula-tion from the previous generation before the non-

dominated sorting procedure is repeated to gener-ate the parent population for the next generation.The algorithm is elitist since it preserves the par-ent population and compares it to the child popu-lation for each successive generation, so the bestdesigns are not lost if the children do not performbetter than the parents.

As initially formulated, NSGA-II does nothandle constraints. There are a number of meth-ods for handling constraints with GAs that wereconsidered. The first method is to either "repair"designs which violate constraints so that they be-come feasible, or to eliminate them from the pop-ulation and replace them with a new random de-sign. This method was not considered due tothe large number of constraints in an aircraft de-sign problem and the low probability of havingrandomly generated feasible designs during theinitial generations. The next method consideredadds the constraint violations to the objectivefunctions with penalty parameters. This methodwas not selected because it requires hand tun-ing to select the penalty parameters for each con-straint and objective function pair. If the penaltyparameters are not large enough there is the pos-sibility of converging to infeasible solutions.

The final method is the constrained tourna-ment selection method [23]. This method modi-fies the non-dominated sorting procedure. Whencomparing two feasible designs, the normal dom-ination comparison operator is used. If a feasi-ble design is compared to an infeasible design,the feasible one is dominant. If two infeasibledesigns are compared, then the design with thesmaller total normalized constraint violation isdominant. It should be noted that there are someflaws with this method. For example, a good de-sign that violates the takeoff constraint by onefoot will be ranked worse than a design withtwice the cost and emissions. There is on goingwork by the authors and others on how to betterhandle constraints in population based methods.The l constrained tournament selection methodwas selected based on its performance on someconstrained test problems. The constraints arenormalized using Eq. (19).

The final modification made to the NSGA-II

8


algorithm was the addition of controlled elitism[23]. Controlled elitism attempts to maintain ageometric distribution in the number of designswith respect to rank. This is beneficial in higherdimensional objective spaces since it maintainsdiversity in the population and discourages pre-mature convergence to a deceptive Pareto front.

The parameters which need to be selected forNSGA-II include the population size, maximumnumber of generations, mutation rate, crossoverrate and the controlled elitism parameter. Also,the variables can be represented as either binarystrings or real-valued variables. A suggested pop-ulation size is at least twenty times the number ofdesign variables, although bigger is always bet-ter. Convergence criteria for MOGAs are diffi-cult to specify, so the algorithms are typically rununtil little change in the non-dominated front isseen. For most problems 500 to 1000 genera-tions are sufficient. The bitwise mutation rate forNSGA-II is typically higher than for single ob-jective GAs and is usually in the range of 0.05 to0.1. The crossover probability used is usually be-tween 0.9 and 1.0. A controlled elitism parametervalue of 0.65 was selected based on suggestionsin Ref. [23].

2.6.2 Integer Programming Solvers

The fleet assignment subspace optimization prob-lem is solved using the CPLEX integer solverin AMPL. The solver can be used for solvingmixed-integer linear programming problems ofthe type outlined for the fleet assignment sub-space. It solves the problem using a branch andcut algorithm. See chapter 7 of Ref. [24] for de-tails.

3 Test Problem Description

In order to test the proposed solution methodol-ogy for the problem formulated in Section 2, atest problem was designed. The test problem in-volves designing a single aircraft type to satisfydemand on a small route network consisting of 4cities and 8 segments. Figure 2 shows the routenetwork with the range and passenger demandon each segment. For a network this small, it is

easy to identify good fleet assignment solutionsfor each capacity in order to verify that the opti-mizer is finding good solutions. The optimal fleetassignment solution should be the same if each ofthe three objectives were optimized individuallyfor such a simple route network with a single air-craft design.

3.1 System Level Aircraft Design Subspace

The system level aircraft design subspace per-forms analyses using PASS and optimizes thesystem using the modified NSGA-II algorithm.The aircraft type is a conventional single-aislecommercial transport with two wing mounted en-gines in the size category of a Boeing 737 or Air-bus A320. The following subsections will discussselection of the design variables and describe theperformance analyses and constraints imposed.

3.1.1 Variables

The variables can be split into four categories:aircraft parameters, engine parameters, operatingconditions, and fleet assignment targets. The air-craft parameters are the maximum takeoff weight(MTOW ), maximum zero fuel weight (MZFW ),capacity (capacity), wing loading (W/S), wingaspect ratio (AR), wing taper ratio (λ), wingquarter chord sweep angle (Λ), wing averagethickness to chord ratio (t/c), horizontal tailarea (SH/SRe f ) and wing location on the fuse-lage (xwing/L). The engine parameters includethe thrust-to-weight ratio (T/W ), overall pres-sure ratio (OPR), fan pressure ratio (FPR) andturbine entry temperature (T ET ). The optimumbypass ratio for minimum specific fuel consump-tion at cruise is computed based on the speci-fied OPR, FPR and T ET using the method inRef. [18]. The two operating conditions thatneed to be specified are the cruise Mach num-ber (Mcruise) and cruise altitude (hcruise). Thefleet assignment targets are the required numberof flights on each segment ( f lightsi) and the to-tal number of aircraft (aircra f t). Since the fleetassignment solution requires that the same num-ber of aircraft start and end the day at each air-port, there is a limited set of values for f lightsi

9


that will satisfy these constraints. The set offeasible values for f lightsi is generated duringpre-processing and saved in an array. The de-sign variable used is the index of this array ratherthan the eight f lighti variables. This reduced thenumber of combinations for the f lightsi variablesfrom 98(= 43046721) to 2868 and cut the num-ber of bits representing these variables in half.

For the genetic algorithm each variable re-quires an upper bound, lower bound, and bit stinglength. There are a total of eighteen design vari-ables. Table 1 lists the minimum, maximum andnumber of bits for each of the design variables.

Table 1 Minimum, maximum and number of bitsfor each design variable.

Variable Min Max BitsMTOW [lb] 80000 342143 18MZFW [lb] 50000 312143 18Capacity 102 192 4W/S[lb/ f t2] 100 163.5 7AR 5 20.75 6λ 0.2 0.51 5Λ[deg] 0 42.333 7t/c 0.07 0.145 4SH/Sre f 0.1 0.41 5xwing/L 0.28 0.59 5T/W 0.2 0.515 6OPR 15 57 6FPR 1.6 2.22 5T ET [R] 2450 3200 4Mcruise 0.7 0.91 6hcruise[ f t] 30000 37000 3f lightsindex 1 2868 12aircra f t 5 20 4

3.1.2 Performance Analysis

The performance characteristics of the aircraftdesign are determined using PASS. The manu-facturer’s empty weight (MEW ) can be computedbased on the MTOW , MZFW and geometry us-ing the weight model within PASS. To determinethe fuel required for each segment, the aircraftis analyzed over the reserve mission shown in

Fig. 3. The reserve mission follows the IATAinternational reserve requirements. For this testproblem, the reserve requirements are probablyexcessive for the ranges specified, but were usedregardless. The aircraft is then analyzed over theactual mission range without the reserve require-ments to determine the fuel burn and NOX emis-sions. For problems with a large number of seg-ments, the fuel burn and NOX emissions can beanalyzed for only a few missions and curve fitto decrease computation time. Noise at the threecertification points is estimated using ICAO rules[12]. 2006 is used as the reference year for fuelcosts ($1.85/gal) and inflation.

3.1.3 Aircraft Design Constraints

Some constraints are performance and certifi-cation driven while others are required to en-sure feasible designs based on the selected de-sign variables for the problem. The last setcould potentially be eliminated by adding itera-tion loops within the analysis, however, the op-timizer should drive these constraints to nearequality, so analysis iteration is not necessarilyrequired.

The balanced field length at takeoff is con-strained to be less than 8000 ft and the land-ing field length less than 7000 ft. The 2nd seg-ment climb gradient with one engine out mustexceed 0.024 to meet FAA regulations for a twoengined aircraft. The required maximum takeoffweight (MZFW+ f uelmax) for all of the missionsmust not exceed the specified maximum take-off weight. The required fuel volume for eachmission must not exceed the available volume inthe main wing box. The required fuel weightmust not exceed the maximum fuel weight. Thewing must be located along the fuselage suchthat the main landing gear attachment point lieswithin the main wing box. The maximum pay-load weight available (MZFW -MEW ) must ex-ceed the specified payload weight correspondingto 225 lb/passenger including luggage, plus anadditional 15% for freight. Stage 4 noise require-ments must be met. The minimum static marginmust be at least 0.05. The thrust available during

10


cruise must exceed the drag by at least 14%. Thehorizontal tail CL cannot exceed its CLmax at anyoperating condition. Similarly, the wing CL mustnot exceed its CLmax at any operating condition.Finally, the wing span is not allowed to exceed262 ft.

3.2 Fleet Assignment Subspace

For the test problem, there were no constraintson basing or on the number of aircraft allowed oneach feasible route. It was assumed that the us-able day was 18 hours long (6 am until midnight)and that layovers at each airport were 1 hour long.There are a total of 202 possible aircraft routesthat can be flown for this simple route networkif the aircraft flies at Mach 0.91 and 30,000 ft,which corresponds to the fastest true airspeed theairplane could fly at.

4 Test Problem Results

Based on the results of many test runs, it was ob-served that the algorithm had a difficult time find-ing the optimum aircraft capacity. This occurredbecause it is much easier to find feasible designsthat meet the takeoff and climb constraints for thesmaller capacities. The GA would find feasiblesolutions for these capacities first which wouldquickly drive the population to capacities lessthan 150 passengers. It was known this was notcorrect since better designs could be found in allobjectives with a capacity of 168 or 180 passen-gers through hand tuning.

To overcome this issue, the GA was run for100 generations for each of the 16 allowed ca-pacities. The 78 best designs were selected fromeach of these sub-populations and used as seeddesigns for the final run of the GA. To dramati-cally speed up computation, a fixed fleet assign-ment solution was used for these seed generationruns, corresponding to the number of flights re-quired to meet demand for each city pair. Theprojections of the seed designs into the planesformed by each pair of objectives are shown inFig. 4. The best performing designs have capac-ities of 180, 168, 144 and 150 passengers. We

expect these designs to proliferate in the final op-timization.

The optimization algorithm was run on a 2GHz laptop PC with 1 GB of RAM for 500 gen-erations with a total computation time of about 38hours. Eight of these hours were for generatingthe seed designs. The population size used was1248. The mutation probability was 0.05, andthe crossover probability was 1.0. Each functionevaluation took approximately one sixth of a sec-ond including the fleet assignment subspace op-timization and about 0.014 seconds with a fixedfleet assignment solution during seed generation.

4.1 Final Pareto Front

The final Pareto front resulting from the opti-mization is shown in Fig. 5. Also included areprojections into the planes formed by each setof two objectives of the initial seed designs andthe Pareto front at generations 200 and 500. In-termediate generations are not shown for clarity.Between generations 200 and 500 there is littlemovement in the Pareto front indicating the al-gorithm has converged. Estimates of the perfor-mance of a few current single aisle aircraft pro-duced by Boeing (737-700, -800 and -900) andAirbus (A319, A320 and A321)are also includedin the plots for comparison. These aircraft weremodeled and analyzed using the same analysiscode and objective functions.

Based on these plots we can see that the algo-rithm progressed to a fairly well-defined Paretofront that forms a curve in the objective space andthat it distributed designs well along the front.From the projected views of the final Paretofronts we see that there are definite trade offs be-tween operating costs and NOX emissions as wellas CO2 emissions and NOX emissions. In gen-eral, reducing CO2 emissions also reduces oper-ating costs. The few designs that are at muchhigher operating cost have a non-optimal num-ber of aircraft for the network, but slightly lowerCO2 and NOX emissions than some designs onthe remainder of the Pareto front. If the numberof aircraft were reduced these would slide into thearea of the Pareto surface where the majority of

11


the designs reside. For a fixed level of NOX emis-sions there is a small trade off between operatingcosts and CO2 emissions, however the width ofthe front in these objectives is small due to thestrong relationship between operating costs andfuel burn. With the increasing cost of fuel overthe value used in this studies we would expectthe width to be even smaller today.

4.2 Aircraft Design Results

In the final Pareto front most of the aircraft de-signs are very similar. The aircraft parameters,operating conditions and fleet assignment valuesall converged to small ranges. Almost all of thevariation over the Pareto front is due to variationsin the engine parameters. Table 2 shows the meanand standard deviation for the variables over thefinal Pareto front. Figure 6 shows histograms foreach of the design variables for the final Paretofront and a top view of the "mean" airplane de-sign. The optimal aircraft capacity turned outto be 168 passengers, one of the values expectedfrom the seed designs.

The variables at or near their bounds arehcruise, T ET , λ and t/c. The allowable cruisealtitude range was selected somewhat arbitrarily,if the optimization were to be re-run the upperbound would probably be extended to a higheraltitude. The turbine entry temperature upperbound was selected at a reasonable turbine mate-rial temperature. Today’s materials may be ableto handle higher temperatures, but reliable dataon the turbine entry temperatures of current com-mercial engines was unavailable. The taper ratiobound was set due to limitations of the analysiscode to properly handle constraints that becomeactive as the taper ratio approaches zero. Thebound on wing thickness was set at what seemedto be a reasonable value based on root sections oftransport wings. The high thickness and higherthan expected wing sweep are the result of theanalysis code. The trades between sweep andthickness for a given mach number are fairly flat,but higher sweeps tend to perform slightly betterleading the optimizer to thicker, high sweep wingdesigns.

The constraints with the smallest margins(active constraints) for most designs in the finalPareto front were the takeoff field length, sec-ond segment climb gradient, maximum takeoffweight, payload weight and either the fuel weightor fuel volume constraints.

Table 2 Mean and standard deviation of variablesover the final Pareto front.

Variable Mean Standard DeviationMTOW [lb] 194690 17.2MZFW [lb] 148330 174Capacity 168 0W/S [lb/ f t2] 140.1 0.334AR 12.6 0.228λ 0.20 0.0Λ [deg] 35.2 0.5t/c 0.145 0.0SH/Sre f 0.27 0.001xwing/L 0.366 0.01T/W 0.32 0.002OPR 23.02 7.98FPR 1.67 0.021T ET [R] 3197 12.0Mcruise 0.79 0.004hcruise [ f t] 36826 383

Figure 7 shows how the engine parametersvary over the Pareto front. As expected the lowNOX designs have lower overall pressure ratios.The lowest CO2 designs have a high turbine en-try temperature, a high overall pressure ratio, anda low fan pressure ratio, which corresponds toa high bypass ratio and high propulsive efficien-cies. Almost all of the designs have higher cruisebypass ratios than engines in service today. Forreference, the CFM56 series of engines used onthe B737 and A320 series of aircraft have bypassratios of about 6 and overall pressure ratios be-tween 21.6 and 32.8, depending on the model.

4.3 Fleet Assignment Results

The final fleet assignment solution was consistentthroughout the final Pareto front. All of the pop-ulation members converged to having the mini-

12


mum number of flights per day on each segmentto meet demand. Most used seven aircraft to sat-isfy the required demand. The few outliers at ahigher DOC + I in the final Pareto surface hadmore aircraft than were necessary. Table 3 showsthe routes flown by the seven aircraft, their uti-lization, and the number of aircraft flying eachroute. The utilization is the sum of the blockhours, but not including any layover time be-tween flights. Table 4 shows the number of flightsand the average load factors on each segment.Both the load factors and utilizations are high in-dicating that the fleet assignment solution is op-timal for the specific aircraft capacity and cruisespeed.

Table 3 Utilization and number of aircraft flyingeach route.

Route Utilization [hr] # of AircraftA.B.C.B.C 14.56 2A.D.A.D 15.72 1C.B.A.D 11.53 1

C.B.C.B.A 14.56 1D.A.D.A 15.72 1

D.C.D.C.D.A 12.07 1Average 14.10 –

Table 4 Number of flights and load factors oneach segment.

Segment Number of Flights Load FactorAB 2 0.685AD 4 0.978BA 2 0.673BC 5 0.851CB 5 0.856CD 2 0.872DA 4 0.939DC 2 0.884

Average – 0.862

5 Conclusions and Future Work

Minimizing operating costs has always been agoal for aircraft designers, but environmental per-formance is quickly becoming a major design fo-cus. In this paper we outline a methodology todesign one or more aircraft types to satisfy de-mand on a given route network to minimize bothoperating costs and emissions. A hierarchical de-composition is used with discipline specific op-timization algorithms. A modified version ofthe NSGA-II MOGA implementing constrainedtournament selection and controlled elitism isused in the system level aircraft design subspace.The CPLEX integer solver is used in the fleet as-signment subspace.

Results are presented for a simple test prob-lem with a 4 city and 8 segment route networkand a single aircraft design. The aircraft pa-rameters, operating conditions and fleet assign-ment solutions remained nearly constant over thePareto Surface, with the surface defined primar-ily by variations in the engine parameters. Theresults show a trade between operating costs andNOX emissions as well as CO2 and NOX emis-sions. Operating costs and CO2 emissions appeardirectly related, with little if any trade off evident.All designs show benefits from using higher by-pass ratio engines and thicker, higher aspect ratiowings than today’s aircraft of the same class.

The methodology presented can be used tosolve many problems more difficult than the sim-ple test problem presented. However, reducingthe computational time for the optimization is animportant consideration. Fortunately, the geneticalgorithm is parallelizable, which can greatly re-duce computation times if more than one proces-sor is used. The computation time will decreasealmost linearly with the increase in the number ofprocessors. This will also allow larger populationsizes to be run for more generations, improvingPareto front performance and confidence in theconverged designs.

In addition to further study of the simple testproblem, possible future work includes a num-ber of modifications and extensions. A prob-lem of immediate interest is the design of two

13


or three aircraft types for more complicated routenetworks. Current aircraft designs could also beincorporated to see how much improvement newdesigns will provide in operating costs and emis-sions. The methodology presented is also wellsuited to designing a family of aircraft. Fuselagestretches would be easy to incorporate, adding afew variables to the system level aircraft designsubspace while giving the fleet assignment sub-space more degrees of freedom. Another pos-sibility is including origin-destination passengerdemand instead of a fixed demand on each seg-ment. In this case, the fleet assignment subspacewould decide which segments should be flownand how passengers should be routed. It wouldbe interesting to see if hub-and-spoke or point-to-point networks evolved to meet the differentobjectives. Further down the road, probabilisticmodels with varying demand on the network andthe effects of winds and weather would be of in-terest in developing robust fleet assignment solu-tions. Another approach that needs to be con-sidered in parallel is to increase the fidelity ofmany of the analyses. A better engine model andNOX model are desired as well as including otheremissions models such as contrail formation andnoise exposure around the airports.

Acknowledgments

The authors acknowledge Steve Altus at Jeppe-sen for discussions related to fleet assignment,air traffic management and environmental issues.The PASS analysis code was developed and pro-vided by Desktop Aeronautics, Inc. The au-thors also acknowledge the contributions fromcolleagues at Stanford University. The first au-thor would also like to thank the National ScienceFoundation for fellowship funding that made thiswork possible.

References

[1] Penner, J. et al., Aviation and the Global At-mosphere, Cambridge University Press Cam-bridge, 1999.

[2] Brooker, P., “Civil Aircraft Design Priorities:

Air Quality? Climate Change? Noise?” TheAeronautical Journal, Vol. 110, No. 1110, 2006,pp. 517–532.

[3] Green, J. et al., “Air Travel-Greener by Design.Mitigating the environmental impact of avia-tion: opportunities and priorities,” The Aero-nautical Journal, Vol. 109, No. 1099, 2005,pp. 361–418.

[4] Houghton, J., Climate Change 1995: The Sci-ence of Climate Change, Cambridge UniversityPress, 1996.

[5] Rogers, H., Lee, D., Raper, D., De F. Foster,P., Wilson, C., and Newton, P., “The impacts ofaviation on the atmosphere,” Aeronautical Jour-nal, Vol. 106, No. 1064, 2002, pp. 521–546.

[6] Panel on Atmospheric Effects of Aviation andBoard on Atmospheric Sciences & Climate andCommission on Geosciences & Environment& Resources and National Research Council,Atmospheric Effects of Aviation: A Review ofNASA’s Subsonic Assessment Project, 1999.

[7] Sausen, R., Isaksen, I., Grewe, V., Hauglustaine,D., Lee, D., Myhre, G., Köhler, M., Pitari, G.,Schumann, U., Stordal, F., et al., “Aviation ra-diative forcing in 2000: An update on IPCC(1999),” Meteorologische Zeitschrift, Vol. 14,No. 4, 2005, pp. 555–561.

[8] Antoine, N. and Kroo, I., “Aircraft Optimizationfor Minimal Environmental Impact,” Journal ofAircraft, Vol. 41, No. 4, 2004, pp. 790–797.

[9] Markish, J. and Willcox, K., “Value-Based Mul-tidisciplinary Techniques for Commercial Air-craft System Design.” AIAA Journal, Vol. 41,No. 10, 2003, pp. 2004–2012.

[10] Liebeck, R. and Center, L. R., Advanced Sub-sonic Airplane Design & Economic Studies, Na-tional Aeronautics and Space Administration,Lewis Research Center; National Technical In-formation Service, distributor, 1995.

[11] Crossley, W. and Mane, M., “System of SystemsInspired Aircraft Sizing Applied to CommercialAircraft/ Airline Problems,” AIAA 5th Aviation,Technology, Integration, and Operations Con-ference(ATIO), 2005, pp. 1–12.

[12] ICAO, E., “Volume 1–Aircraft Noise, 4th Ed.”2005.

[13] ICAO, E., “Volume 2–Aircraft Engine Emis-sions, 2nd Ed.” 1993.

14


[14] ICAO, E., “Environmental Report 2007,” 2007.[15] Schumann, U., “Formation, Properties and Cli-

matic Effects of Contrails,” Comptes rendus-Physique, Vol. 6, No. 4-5, 2005, pp. 549–565.

[16] Norris, G. and Wagner, M., Airbus A380: Su-perjumbo of the 21st Century, Zenith Press,Osceola, WI, USA, 2005.

[17] Kroo, I., “An Interactive System for AircraftDesign and Optimization,” Aiaa paper 92–1190, AIAA Aerospace Design Conference,Feb. 1992.

[18] Mattingly, J., Heiser, W., and Pratt, D., AircraftEngine Design, Aiaa, 2002.

[19] Torenbeek, E., “Synthesis of Subsonic AircraftDesign,” Student edition, Delft University ofTechnology/Martinus Nijhoff , 1982.

[20] ICAO, I., “Engine Exhaust Emissions DataBank.–2007,” Tech. rep., Doc 9646-AN/943. In-ternational Civil Aviation Organisation, Mon-treal, Canada, 2007.

[21] Norman, P., Lister, D., Lecht, M., Madden,P., Park, K., Penanhoat, O., Plaisance, C., andRenger, K., “Development of the technical ba-sis for a New Emissions Parameter coveringthe whole AIRcraft operation: NEPAIR, FinalTechnical Report,” Tech. rep., CAEP/6-IP/17Appendix B, 2003.

[22] Fourer, R., Gay, D., and Kernighan, B., AMPL:A Modeling Language for Mathematical Pro-gramming, Thomson/Brooks/Cole, 2003.

[23] Deb, K., Multi-Objective Optimization UsingEvolutionary Algorithms, Wiley, 2001.

[24] CPLEX, I., “10.0 User’s guide,” ILOG, Moun-tain View, CA, 2006.

[25] Deb, K., Agrawal, S., Pratap, A., and Meyari-van, T., “A Fast Elitist Non-Dominated Sort-ing Genetic Algorithm for Multi-Objective Op-timization: NSGA-II,” Proceedings of the Par-allel Problem Solving from Nature VI Confer-ence, 2000, pp. 849–858.

[26] Zitzler, E., Laumanns, M., Thiele, L., et al.,“SPEA2: Improving the Strength Pareto Evolu-tionary Algorithm,” EUROGEN, 2001, pp. 95–100.

15


16


(a) Takeoff (b) Climb

(c) Approach (d) Idle

Fig. 1 Fits of NOX emission index at the four certification points as a function of overall pressure ratio.

17


Fig. 2 Route Network for test problem.

18


Fig. 3 Reserve mission profile.

19


(a) CO2 vs. DOC + I

(b) NOX vs. CO2

(c) NOX vs. DOC + I

Fig. 4 Seed designs for the final optimization run.

20


(a) CO2 vs. DOC + I vs. NOX (b) CO2 vs. DOC + I

(c) NOX vs. DOC + I (d) NOX vs. CO2

Fig. 5 The final Pareto front and the projections into the objective planes of the population seeds, thePareto front at generations 200 and 500, and the estimated performance of current aircraft.

21


(a) MTOW (b) MZFW (c) capacity

(d) W/S (e) AR (f) λ

(g) Λ (h) t/c (i) SH/Sre f

(j) xwing/L (k) T/W (l) OPR

(m) FPR (n) T ET (o) Mcruise

(p) hcruise (q) Mean Airplane

Fig. 6 Histograms of each airplane design variable and a top view of the mean airplane from the finalPareto front.

22


Fig. 7 Engine parameter variation over the final Pareto front.

23

Documents

MULTI-OBJECTIVE AIRCRAFT OPTIMIZATION FOR MINIMUM …aero.stanford.edu/Reports/multiobjectiveaircraftandroute... · · 2008-10-01MULTI-OBJECTIVE AIRCRAFT OPTIMIZATION FOR MINIMUM