1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Multidisciplinary System Multidisciplinary System Design Optimization (MSDO)Design Optimization (MSDO)
Multiobjective Optimization (I)Lecture 16
31 March 2004
byProf. Olivier de Weck
2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Where in Framework ?Where in Framework ?
Discipline A Discipline B
Discipline C
Inpu
t
Out
put
TradespaceExploration
(DOE)
Optimization Algorithms
Numerical Techniques(direct and penalty methods)
Heuristic Techniques(SA,GA, Tabu Search)
1
2
n
x
x
x
Coupling
1
2
z
J
J
J
ApproximationMethods
Coupling
Sensitivity Analysis
MultiobjectiveOptimization
Isoperformance
Objective Vector
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Lecture ContentLecture Content
• Why multiobjective optimization?• Example – twin peaks optimization• History of multiobjective optimization• Weighted Sum Approach (Convex Combination)• Dominance and Pareto-Optimality• Pareto Front Computation - NBI
4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Multiobjective Optimization Problem Multiobjective Optimization Problem Formal DefinitionFormal Definition
( )
, , 1, ..., )
min ,
s.t. , 0
, =0
(i LB i i UB i nx x x =
≤
≤ ≤
Design problem may be formulatedas a problem of Nonlinear Programming (NLP). WhenMultiple objectives (criteria) are present we have a MONLP
( ) ( )[ ]
1
2
1
1
1
1
where
( ) ( )
( ) ( )
=
=
=
=
T
z
T
i n
T
m
T
m
J J
x x x
g g
h h
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Multiple ObjectivesMultiple Objectives
1
2
3
cost [$]
- range [km]
weight [kg]
- data rate [bps]
- ROI [%]
i
z
J
J
J
J
J
= =
The objective can be a vector J of z system responsesor characteristics we are trying to maximize or minimize
Often the objective is ascalar function, but forreal systems often we attempt multi-objectiveoptimization:
Objectives are usuallyconflicting.
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Why multiobjective optimization ?Why multiobjective optimization ?
While multidisciplinary design can be associated with the traditional disciplines such as aerodynamics, propulsion, structures, and controls there are also the lifecycle areas ofmanufacturability, supportability, and cost which require consideration.
After all, it is the balanced design with equal or weighted treatment of performance, cost, manufacturability and supportability which has to be the ultimate goal of multidisciplinary system design optimization.
Design attempts to satisfy multiple, possiblyconflicting objectives at once.
7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Example F/AExample F/A--18 Aircraft18 AircraftDesign
DecisionsObjectives
Aspect RatioDihedral AngleVertical Tail AreaEngine Thrust Skin Thickness
# of EnginesFuselage SplicesSuspension PointsLocation of MissionComputerAccess Door Locations
SpeedRangePayload CapabilityRadar Cross SectionStall SpeedStowed Volume
Acquisition costCost/Flight hourMTBFEngine swap timeAssembly hours
Avionics growthPotential
8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Multiobjective ExamplesMultiobjective Examples
Production Planningmax {total net revenue}max {min net revenue in any time period}min {backorders}min {overtime}min {finished goods inventory}
Aircraft Designmax {range}max {passenger volume}max {payload mass}min {specific fuel consumption}max {cruise speed}min {lifecycle cost}
1
2
z
J
J
J
=DesignOptimization
OperationsResearch
9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Multiobjective vs. MultidisciplinaryMultiobjective vs. Multidisciplinary
• Multiobjective Optimization– Optimizing conflicting objectives– e.g., Cost, Mass, Deformation – Issues: Form Objective Function that represents designer
preference! Methods used to date are largely primitive.
• Multidisciplinary Design Optimization– Optimization involves several disciplines– e.g. Structures, Control, Aero, Manufacturing– Issues: Human and computational infrastructure, cultural,
administrative, communication, software, computing time, methods
• All optimization is (or should be) multiobjective– Minimizing mass alone, as is often done, is problematic
10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Multidisciplinary vs. MultiobjectiveMultidisciplinary vs. Multiobjective
single discipline multiple disciplines
sing
le o
bjec
tive
mul
tiple
obj
.
single discipline multiple disciplines
Minimize displacements.t. mass and loading constraint
F
δl
mcantilever beam support bracket
Minimize stamping costs (mfg) subject
to loading and geometryconstraint
F
D
$
airfoilα (x,y)
Maximize CL/CD and maximizewing fuel volume for specified α, vo
Vfuelvo
Minimize SFC and maximize cruisespeed s.t. fixed range and payload
commercial aircraft
Image taken from NASA's website. http://www.nasa.gov.
11 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Example: Double Peaks OptimizationExample: Double Peaks OptimizationObjective: max J= [ J1 J2]T (demo)
( )
( )
2 21 2
2 21 2
2 21 2
2 ( 1)1 1
3 511 2
( 2)1 2
3 1
105
3 0.5 2
x x
x x
x x
J x e
xx x e
e x x
− − +
− −
− + −
= −
− − −
− + +
( ) 2 22 1
2 2 2 22 1 2 1
2 ( 1)2 2
(2 )3 522 1
3 1
10 35
x x
x x x x
J x e
xx x e e
− +
− − − −
= +
− − + − −
12 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Double peaks optimizationDouble peaks optimization
Optimum for J1 alone: Optimum for J2 alone:
x1* =0.05321.5973
J1* = 8.9280
J2(x1*)= -4.8202
x2* =-1.58080.0095
J1(x2*)= -6.4858
J2* = 8.1118
Each point x1* and x2* optimizes objectives J1 and J2 individually.Unfortunately, at these points the other objective exhibits a low objective function value. There is no single point that simultaneouslyoptimizes both objectives J1 and J2 !
13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Tradeoff between Tradeoff between JJ11 andand JJ22
• Want to do well with respect to both J1 and J2
• Define new objective function: Jtot=J1 + J2
• Optimize Jtot
Result:
Xtot* =0.87310.5664
J(xtot*) =3.0173 J13.1267 J2
Jtot* = 6.1439
=
max(J1)
max(J2)
tradeoffsolution
max(J1+J2)
14 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
History (1) History (1) –– Multicriteria Decision MakingMulticriteria Decision Making
• Life is about making decisions. Most people attempt to make the “best” decision within a specified set of possible decisions.
• Historically, “best” was defined differently in different fields:– Life Sciences: The best referred to the decision that minimized or
maximized a single criterion.– Economics: The best referred to the decision that simultaneously
optimized several criteria.
• In 1881, King’s College (London) and later Oxford Economics Professor F.Y. Edgeworth is the first to define an optimum for multicriteria economic decision making. He does so for the multiutility problem within the context of two consumers, P and π:
– “It is required to find a point (x,y,) such that in whatever direction we take an infinitely small step, P and π do not increase together but that, while one increases, the other decreases.”
– Reference: Edgeworth, F.Y., Mathematical Psychics,P. Keagan, London, England, 1881.
15 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
History (2) History (2) –– Vilfredo ParetoVilfredo Pareto• Born in Paris in 1848 to a French Mother and Genovese
Father• Graduates from the University of Turin in 1870 with a
degree in Civil Engineering– Thesis Title: “The Fundamental Principles of Equilibrium in
Solid Bodies”
• While working in Florence as a Civil Engineer from 1870-1893, Pareto takes up the study of philosophy and politics and is one of the first to analyze economic problems with mathematical tools.
• In 1893, Pareto becomes the Chair of Political Economy at the University of Lausanne in Switzerland, where he creates his two most famous theories:
– Circulation of the Elites– The Pareto Optimum
• “The optimum allocation of the resources of a society is not attained so long as it is possible to make at least one individual better off in his own estimation while keeping others as well off as before in their own estimation.”
• Reference: Pareto, V., Manuale di Economia Politica, SocietaEditrice Libraria, Milano, Italy, 1906. Translated into English by A.S. Schwier as Manual of Political Economy, Macmillan, New York, 1971.
16 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
History (3) History (3) –– Extension to EngineeringExtension to Engineering
• After the translation of Pareto’s Manual of Political Economyinto English, Prof. Wolfram Stadler of San Francisco State University begins to apply the notion of Pareto Optimality to the fields of engineering and science in the middle 1970’s.
• The applications of multi-objective optimization in engineering design grew over the following decades.
• References:– Stadler, W., “A Survey of Multicriteria Optimization, or the Vector
Maximum Problem,” Journal of Optimization Theory and Applications, Vol. 29, pp. 1-52, 1979.
– Stadler, W. “Applications of Multicriteria Optimization in Engineering and the Sciences (A Survey),” Multiple Criteria Decision Making –Past Decade and Future Trends, ed. M. Zeleny, JAI Press, Greenwich, Connecticut, 1984.
– Ralph E. Steuer, “Multicriteria Optimization - Theory, Computation and Application”, 1985
17 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Notation and ClassificationNotation and Classification
Ref: Ralph E. Steuer, “Multicriteria Optimization - Theory, Computation and Application”, 1985
Traditionally - single objective constrained optimization:
( )max
. .
f
s t S
=∈
( ) objective function
feasible region
f J
S
If f(x) linear & constraints linear & single objective = LPIf f(x) linear & constraints linear & multiple obj. = MOLPIf f(x) and/or constraints nonlinear & single obj.= NLPIf f(x) and/or constraints nonlinear & multiple obj.= MONLP
18 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Mapping SetsMapping Sets
Linear
Nonlinear
Decision Space(Design Space)
Criterion Space(Objective Space)
x1
x2
x1
x2
J1
J2
S Z
c1
c2
x1
x4
x3x2
J1
J3
J4
J2
J1
J2
Z’
J1
J4
J2x1
MOLP
MONLP
x2
x3
x4S’
S’’ J3
Z’’
Z’’’
Set of images ofall points in set S
19© Massachusetts Institute ofTechnology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Map: Double Peaks Map: Double Peaks –– manymany--toto--oneone
Xtot*
J(xtot*)
A function f which may (but does not necessarily) associate a given member of the range of f with more than one member of the domain of f.
“Range”“Domain”
A
Domain Range
Bfa
Many-To-One
20 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Formal Solution of a MOO ProblemFormal Solution of a MOO ProblemTrivial Case:There is a point that simultaneously optimizesall objectives
*x S∈, where 1iJ i z≤ ≤
Such a point almost never exists - i.e. there is no pointthat will simultaneously optimizes all objectives at once
Two fundamental approaches:
( ){ }( )
1 2max , , ,
. . 1 i z
z
i i
U J J J
s t J f
S
= ≤ ≤∈
Scalarization Approaches
ParetoApproaches
1 2
1 2
and for
at least one
i i
i i
J J i
J J
i
≥ ∀
>
Preferences included upfront
Preferences included a posteriori
21 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
NotationNotation
SOLPSOLP { }max T J S= ∈
S feasible region in decision spaceif S is determined by linear constraints
{ }, 0,n mS = ∈ = ≥ ∈
MOLPMOLP { }
{ }
11max
max . .
T
zTz
J
J s t S
=
= ∈Multiobjectivelinear program
Single objectivelinear program
22 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Notation (II)Notation (II)
C z x n Criterion matrix1 2 zc c c=
Gradientvector of z objectives
1 2
n
Ti i inx x x
∈
=
point in decision space
[ ]1 2
z
T
zJ J J
∈
=
(Criterion) Objective vector
23 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
SOLP versus MOLPSOLP versus MOLP
x1
x2
S
c1
x1*
x2*c2constraints
Optimal solutionif only J1 considered
Optimal solutionif only J2 considered
What is the optimalsolution of a MOLP?{ }
{ }1
2
max
max
. . S
J
J
s t
=
=
∈
24 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
WeightedWeighted--Sum ApproachSum Approach
x1
x2
Sc1
x1
x2
c2
constraints
Each objective i is multiplied by a strictly positive scalar λi
1
0, 1k
zi i
i
λ λ=
Λ = ∈ > =
x3*Solve thecomposite orWSLP:
{ }max S∈c31 2λ λ=
Strictly convexcombinationof objectives
criterioncone
25 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Weighted Sum: Double PeaksWeighted Sum: Double Peaks( )1 21 where [0,1]totJ J Jλ λ λ= + − ∈
Demo: At each setting of λ we solve a new singleobjective optimization problem – the underlyingfunction changes at each increment of λ
∆λ=0.05
26 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Weighted Sum Approach (II)Weighted Sum Approach (II)
λ=1
λ=0
Weighted sumfinds interesting, solutions but missesmany solutions ofinterest.
27 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Weighted Sum (WS) ApproachWeighted Sum (WS) Approach
1
zi
MO ii i
J Jsf
λ=
=
Max(J1 /sf1)
Min(J2 /sf2)
miss thisconcave region
Paretofront
• convert back to SOP• LP in J-space• easy to implement• scaling important !• weighting determines which point along PF is found• misses concave PF
λ2>λ1
λ1>λ2
J-hyperplaneJ*i
J*i+1
PF=Pareto Front(ier)
weight
scalefactor
28 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
What are the weights What are the weights λλλλλλλλ ??
: zU →Decision Maker has a utility function U=f(J), J=J(x)
Then:
1
z
x x ii i
UU J
J=
∂∇ = ∇∂
Chain rule
Gradient of U at x S∈
where
i
i
x i
i
n
J
x
J
J
x
∂∂
∇ =∂∂
For LP’s1
z
ii
w=1
z
xi i
UU
J=
∂∇ =∂ Direction of
Gradient at x
1
ii
U Jw
U J
∂ ∂=∂ ∂
1
ii z
jj
w
wλ
=
=Weights determinedirection of gradientvector of U normalize
29 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Group Exercise: Weights (5 min)Group Exercise: Weights (5 min)We are trying to build the “optimal” automobile
There are six consumer groups:-G1: “25 year old single” (Cannes, France)
-G2: “family w/3 kids” (St. Louis, MO)
-G3: “electrician/entrepreneur” (Boston, MA)
-G4: “traveling salesman” (Montana, MT)
-G5: “old lady” (Rome, Italy)
-G6: “taxi driver” (Hong Kong, China)
Objective Vector:J1: Turning Radius [m]J2: Acceleration [0-60mph]J3: Cargo Space [m3]J4: Fuel Efficiency [mpg]J5: Styling [Rating 0-10]J6: Range [km]J7: Crash Rating [poor-excellent]
J8: Passenger Space [m3]J9: Mean Time to Failure [km]
Assignment: Determine λi , i =1…99
1
1000ii
λ=
=WB=wheelbase, ED=engine displacement
x=[WB ED…]T
( )f=
30 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
What are the scale factors What are the scale factors sfsfii??
• Scaling is critical in multiobjective optimization• Scale each objective by sfi:• Common practice is to scale by sfi = Ji*• Alternatively, scale to initial guess J(xo)=[1..1]T
• Multiobjective optimization then takes place in a non-dimensional, unit-less space
• Recover original objective function values by reverse scaling
i i iJ J sf=
Example: J1=range [sm]J2=fuel efficiency [mpg]
sf1=573.5 [sm]sf2=36 [mpg]
“best in GM class”
Saab 9-5
Suzuki “Swift”
31 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Properties of optimal solutionProperties of optimal solution
( ) ( )*optimal if
for and for S
≥
∈ ≠This is why multiobjective optimization is alsosometimes referred to as vector optimization
x* must be an efficient solution
S∈ is efficient if and only if (iff) its objective vector(criteria) J(x) is non-dominated
A point is efficient if it is not possible to move feasiblyfrom it to increase an objective without decreasing at leastone of the others
S∈
(maximization)
32 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Dominance (assuming maximization)Dominance (assuming maximization)
Let be two objective (criterion) vectors.
Then J1 dominates J2 (weakly) iff
Moreprecisely:
z∈1
2i
z
J
JJ
J
= and ≥ ≠
1 2 1 2 and for at least one i i i iJ J i J J i≥ ∀ >
Also J1 strongly dominates J2 iff
Moreprecisely:
>1 2 i iJ J i> ∀
33 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Dominance Dominance -- ExerciseExercise
max{range}min{cost}max{passengers}max{speed}
[km][$/km][-][km/h]
7587 6695 3788 8108 5652 6777 5812 7432321 211 308 278 223 355 401 208112 345 450 88 212 90 185 208950 820 750 999 812 901 788 790
#1 #2 #3 #4 #5 #6 #7 #8
MultiobjectiveAircraft Design
Which designs are non-dominated ? (5 min)
34 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Set TheorySet TheoryNon-dominated Designs: #1, #2, #3, #4 and #8Dominated Designs: #5, #6, #7
Set Theory: DND
D ND∩ = ∅
S∈
J
Z
,D Z ND Z⊂ ⊂
D ND Z∪ =
J*
( ) ND∈
A solution must be feasible
A solution is either dominated or non-dominatedbut cannot be both at the same time
All dominated and non-dominatedsolutions must be feasible
All feasible solutions are either non-dominatedor dominated
Pareto-optimal solutions are non-dominated
Z∈
35 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Procedure (I)Procedure (I)Algorithm for extracting non-dominated solutions:
Pairwise comparison
7587 > 6695321 > 211 112 < 345950 > 820
Score#1
Score#2#1 #2
1 00 10 11 0
2 vs 2Neither #1 nor #2 dominate each other
7587 > 6777321 < 355 112 > 90950 > 901
Score#1
Score#6#1 #6
1 0
1 0
1 01 0
4 vs 0Solution #1 dominatessolution #6
In order to be dominated a solution musthave a”score” of 0 in pairwise comparison
36 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Procedure (II)Procedure (II)for ind1=1:(n-1)
for ind2=(ind1+1):n Ja=Jall(:,ind1).*sign(minmax);Jb=Jall(:,ind2).*sign(minmax); scorea=0;scoreb=0; for indz=1:z
if Ja(indz)>Jb(indz)scorea=scorea+1;
elseif Ja(indz)<Jb(indz)scoreb=scoreb+1;
else% both solutions are equal
endendif scoreb==0&scorea~=0
dmatrix(ind1,ind2)=1; %a dominates belseif scorea==0&scoreb~=0
dmatrix(ind2,ind1)=1; %b dominates aelse
% no domination in this pairend
endend
Pairwisecomparison
PopulateDominationMatrix
paretosort.m
37 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Domination MatrixDomination Matrix
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
12000001
Solution 2 dominatesSolution 5
0 0 0 0 1 1 2 0
Shows which solution dominates which othersolution (horizontal rows) and (vertical columns)
Solution 7 is dominatedby Solutions 2 and 8
Row Σ
j-th row indicateshow manysolutionsj-th solutiondominates
Column Σ
k-th column indicatesby how many solutionsthe k-th solution is dominated
Non-dominated solutions have a zero in the column Σ !
38 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Dominance versus EfficiencyDominance versus Efficiency
Whereas the idea of dominance refers to vectors incriterion space J, the idea of efficiency refers to pointsin decision space x.
Mapping:
( ){ },zZ f S= ∈ = ∈
MOLP
MONLP
{ },zZ S= ∈ = ∈
dominance(objective space)
efficiency(decisionspace)
Z= reachable range in objective space
39 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
EfficiencyEfficiency
( ) ( )( ) ( )
A point is efficient iff there does
not exist another such that
and . Otherwise is inefficient.
S
S
∈
∈ ≥
≠
A point x is efficient if its criterion (objective) vectoris not dominated by the criterion vector of someother point in S.
Efficient set ND
Can use this criterionas a Pareto Filter if thedesign space has beenexplored (e.g. DoE).
40 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Double Peaks: NonDouble Peaks: Non--dominated Setdominated Set
Filtered theFull FactorialSet: 3721
Non-dominatedset approximatesPareto frontier:79 points (2.1%)
41 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
ParetoPareto--Optimal vs NDOptimal vs ND
max (J1)
min(J2)TrueParetoFront
ApproximatedPareto Front
D ND PO
All pareto optimal points are non-dominatedNot all non-dominated points are pareto-optimal
√√√√√√√√√√√√ √√√√
It’s easier to show dominatedness than non-dominatedness !!!
Obtaindifferentpoints fordifferent weights
42 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Direct Pareto Front CalculationDirect Pareto Front Calculation
SOO: find x*MOO: find PF
PFJ1
J2
PFJ1
J2
bad good
- It must have the ability to capture all Pareto points- Scaling mismatch between objective manageable- An even distribution of the input parameters (weights)
should result in an even distribution of solutions
A good method is Normal-Boundary-Intersection (NBI)
43 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Normal Boundary Intersection (I)Normal Boundary Intersection (I)Goal: Generate Pareto points that are well-distributed
1J
2J
10
1
Reachable Range
NU*
1J
*2J
0
uUtopiaPoint
1. Carry out single objective optimization: ( )* 1,2,...,i iJ J i z= ∀ =
2. Find utopia point
( )i i
ii
J JJ
l
−=
( ) ( ) ( )1 2
T
zJ J J=
5. Normalize
3. Find Nadir Point
1 2
TN N NzJ J J=
( ) ( ) ( )minT
Ni i i iJ J J J=
4. Nadir-Utopia Distance
[ ]1 2
T
zl l l= =
1, 2,...,i z=
44 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Normal Boundary Intersection (2)Normal Boundary Intersection (2)
• Carry out a series of optimizations • Restricted design space: U – Utopia Line between
anchor points, NU – normal to Utopia line• Feasible region restricted to one part of Range• Find Pareto point for each NU setting• Move NU from to in even increments• Yields remarkably even distribution of Pareto points• Applies for z>2, U-line becomes a Utopia-hyperplane
*1J *
2J
Reference: Das I. and Dennis J, “Normal-Boundary Intersection:A New Method for Generating Pareto Optimal Points in MulticriteriaOptimization Problems”, SIAM Journal on Optimization, Vol. 8, No.3, 1998, pp. 631-657
45 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Lecture SummaryLecture Summary
• A multiobjective problem has more than one optimal solution• All points on Pareto Front are non-dominated• Methods:
• Weighted Sum Approach (Caution: Scaling !)• Pareto-Filter Approach• Normal Boundary Intersection (NBI) • More methods in the next lecture
The key difference between multiobjective optimizationmethods can be found in how and when designerpreferences are brought into the process.
…. More in next lecture
46 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics
Remember ….Remember ….
Pareto Optimal means …..
“Take from Peter to pay Paul”