1 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Multidisciplinary System
Design Optimization (MSDO)
Multiobjective Optimization (II)
Lecture 17
April 5, 2004
by
Prof. Olivier de Weck
2 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
MOO 2 Lecture Outline
Lecture 2 (today)
? Alternatives to Weighted Sum (WS) Approach
Multiobjective Heuristic Programming
Utility Function Optimization
Physical Programming (Prof. Messac)
Application to Space System Optimization
Lab Preview (Friday 4-9-2003 – Section 1)
3 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Weighted Sum (WS) Approach
1
z
i
MOi
i
i
w
JJ
sf
=
=
|
utopia
Max(J
1
)
Min(J
2
)
miss this
concave region
Pareto
front
convert back to SOP
LP in J-space
easy to implement
scaling important !
weighting determines
which point along PF is
found
misses concave PF
w
2
>w
1
w
1
>w
2
J-hyperplane
J*
i
J*
i+1
4 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Weighted Square Sum Approach
22
11 2 2
J wJ wJ=+
Obj. Fun. Line
J1
J2
Ref: Messac
5 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Compromise Programming (CP)
Obj. Fun. Line
11 2 2
nn
JwJ wJ=+
55
This allows
“access” to the
non-convex part of the
Pareto front
6 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Multiobjective Heuristics
Pareto ranking scheme
Allows ranking of population
without assigning preferences
or weights to individual
objectives
Successive ranking and
removal scheme
Deciding on fitness of
dominated solutions is more
difficult.
Pareto ranking for
a minimization problem.
Pareto Fitness - Ranking
Recall: Multiobjective GA
This number comes
from the D-matrix
7 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Example Multiobjective GA
()
2
11
1
1
,..., 1 exp
n
ni
i
fx x x
n
=
ao
§·
=? ? ?
??
¨?
?1
??
|
Minimization
Objective 1
Objective 2
()
2
11
1
1
,..., 1 exp
n
ni
i
fx x x
n
=
ao
§·
=? ? +
??
¨?
?1
??
|
No mating
restrictions
With mating
restrictions
8 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Double Peaks Example: MO-GA
Multiobjective Genetic Algorithm
Generation 1 Generation 10
9 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Utility Function Approach
Decision maker has utility function
This function might or might not be known mathematically
U maps objective vector to the real line
:
z
U →\MOLP:
MONLP:
()
{ }
max ,US=∈JJ Cxx
() ()
{ }
max ,UfS=∈JJ x x
(0,0)
(0,4)
(3,0)
=
=
=
1
2
3
x
x
x
{ }
{}
112
21
12
12
12
max
max s.t.
4x 3 12
, 0 where
2
J xx
Jx
x
xx
UJJ
=+
=
+≤
≥
=
x
1
x
2
c
1
x
2
c
2
x
3
x
1
S
U=24
U=18
Example:
10 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Utility Function Shapes
J
i
U
i
J
i
U
i
U
i
J
i
U
i
J
i
Monotonic
increasing
decreasing
Strictly
Concave
Convex
Concave
Convex
Non-monotonic
Cook:
Smaller-is-better (SIB)
Larger-is-better (LIB)
Nominal-is
-better (NIB)
Range
-is-better (RIB)
-
Messac:
Class 1S
Class 2S
Class 3S Class 4S
11 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Example: Room Control Optimization
Want: - temperature in ideal range 68-72 oF
- humidity above 56% is undesirable
Assume:
temperature
humidity
T
H
=
=
1
2
cx
cx
U
T
T [oF]
68 72
undesirable
ideal
U
H
H [%]
56
undesirable
ideal
Formulate as a MOLP
{ }
{}
11
2
1
1
2
2
112
min
min s.t.
68
72
56
, , , , 0
dd
d
d
d
d
ddd
?+
+
?
+
+
??+
+
+≥
?≤
?≤
≤≥
1
1
cx
cx
cx
Ax b x
Using
deviational
variables
12 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Aggregated Utility
The total utility becomes the weighted sum of partial utilities:
Attribute J
i
customer 1
customer 2
customer 3
Caution: “Utility” is a surrogate
for “value”, but while “value”
has units of [$], utility is
unitless.
interviews
Combine single utilities
into overall utility function:
Steps: MAUA
1. Identify Critical Objectives/Attrib.
2. Develop Interview Questionnaire
3. Administer Questionnaire
4. Develop Agg. Utility Function
5. Analyze Results
(performance i)
U
i
(J
i
)
ki’s determined during interviews
K is dependent scaling factor
… sometimes called multi-attribute utility analysis (MAUA)
1.0
E.g. two utilities combined:
()
12 12 1 2 1 1 2 2
,()()()()UJ J KkkUJUJ kUJ kUJ=++
For 2 objectives: 1212
(1 ) /K kkkk=??
13 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Notes about Utility Maximization
Utility maximization is very common and well accepted
Usually U is a non-linear combination of objectives J
Physical meaning of aggregate objective is lost (no units)
Need to obtain a mathematical representation for U(J
i
) for
all I to include all components of utility
Utility function can vary drastically depending on decision
maker …e.g. in U.S. Govt change every 3-4 years
14 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Physical Programming
Classify Each Design Objective
SOFT
Class-1S Smaller-Is-Better, i.e. minimization.
Class-2S Larger-Is-Better, i.e. maximization.
Class-3S Value-Is-Better.
Class-4S Range-Is-Better.
HARD
Class-1H Must be smaller.
Class-2H Must be larger.
Class-3H Must be equal.
Class-4H Must be in range.
Ref: Prof. Achille Messac, RPI
15 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Quantify Preference for Each Design Metric
Ex: Mass of Beam
Highly Desirable < 250 (kg)
Desirable 250 - 275
Tolerable 275 - 300
Undesirable 300 - 325
Highly Undesirable 325 - 350
Unacceptable > 350
Physical Programming
16 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Mass
Quantity M
i
nim
ized
Inside Cod
e
Physical Programming
18 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Preference Function of
Each Objective
Cost (preference) is on the
vertical axis, and will be
minimized.
The value of the design metric
(obj) is on the horizontal axis.
The designer chooses limits of
several ranges for each design
metric.
Each range defines relative
levels of desirability within a
given design metric (obj).
We then have a preference
function for each design metric.
These preference functions are
added to form an aggregate
preference function.
GU
S
19 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
μ
i
(x) ≤ v
i5
min
x
P(μ)=
1
n
sc
Σ
i =1
n
sc
P
i
[μ
i
(x)]
(for soft classes)
(for class 1S metrics)
(for class 2S metrics)
(for class 3S metrics)
(for class 4S metrics)
(for class 1H metrics)
(for class 2H metrics)
(for class 3H metrics)
(for class 4H metrics)
(for des. variable. constraints)
Subject to
μ
i
(x) ≥ v
i5
v
i5L
≤ μ
i
(x) ≤ v
i5R
v
i5L
≤ μ
i
(x) ≤ v
i5R
μ
i
(x) ≤ v
i,max
μ
i
(x) ≥ v
i,min
μ
i
(x)=v
i,val
v
i,min
≤ μ
i
(x) ≤ v
i,max
x
j,min
≤ x
j
≤x
j,max
Physical Programming Problem Model
Nomenclature
here μ is used
similar to J
in the class
20 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Application to System Design
Multiobjective Problem:
– Minimize Cost AND Maximize Performance Simultaneously
Which design is best according to these decision criteria?
Key Point: Multi-Objective problems can have more than one solution!
Single objective problems have only one true solution.
0 500 1000 1500 2000
0
0.5
1
1.5
2
Multi-Objective Illustration
Performance (total # images)
L
i
f
e
c
ycl
e C
o
st
(
$
b
i
l
l
i
o
n
s)
Design 3
Design 2
Design 4
Design 1
Design 5
(500,$0.5B)
(1400,$0.8B)
(2000,$1.5B)
(1600,$1.8B)
(1000,$1.3B)
Optimize Architecture of
Terrestrial Planet Finder (TPF)
Mission (expected Launch 2011)
21 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
The Pareto Boundary
In a two-dimensional trade space (I.e. two decision criteria),
the Pareto Optimal set represents the boundary of the most
design efficient solutions.
0 500 1000 1500 2000 2500 3000 3500 4000
800
1000
1200
1400
1600
1800
2000
2200
Performance (total # of images)
Li
f
e
c
ycl
e C
o
s
t
(
$
M
)
TPF System Trade Space Pareto-Optimal Front
Dominated Solutions
Non-Dominated Solutions
$2M/Image
$1M/Image
$0.5M/Image
$0.25M/Image
4AP - 1D SSI - 4m Dia - 1AU
4AP - 1D SCI - 1m Dia - 1AU
SSI
SCI
22 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
TPF Pareto Optimal Set
# “Images” LCC ($B) Orbit (AU) # Apert.’s Architecture Apert.
Diam. (m)
502 0.743 1.5 4 SCI-1D 1
577 0.762 2.0 4 SCI-1D 1
651 0.767 2.5 4 SCI-1D 1
1005 0.768 1.5 4 SCI-1D 2
1114 0.788 2.0 4 SCI-1D 2
1171 0.790 2.5 4 SCI-1D 2
1195 0.807 1.5 6 SCI-1D 2
1292 0.811 1.5 6 SCI-2D 2
1317 0.830 1.5 8 SCI-1D 2
1424 0.836 2.0 4 SCI-1D 3
1426 0.838 1.5 8 SCI-2D 2
1464 0.867 2.5 6 SCI-2D 2
1631 0.877 1.5 6 SCI-1D 3
1684 0.881 1.5 6 SCI-2D 3
1687 0.932 2.0 6 SCI-1D 3
1828 0.936 2.0 6 SCI-2D 3
1881 0.980 1.5 8 SCI-2D 3
1978 0.982 1.5 6 SCI-1D 4
2035 1.086 2.0 8 SCI-2D 3
2132 1.112 1.5 8 SCI-1D 4
2285 1.120 1.5 8 SCI-2D 4
2328 1.190 2.5 6 SCI-2D 4
2398 1197 3.0 6 SCI-2D 4
2433 1.212 4.0 6 SCI-2D 4
2472 1.221 4.5 6 SCI-2D 4
2482 1.227 5.0 6 SCI-2D 4
2487 1.232 5.5 6 SCI-2D 4
2634 1.273 2.5 8 SCI-2D 4
2700 1.280 3.0 8 SCI-2D 4
2739 1.288 3.5 8 SCI-2D 4
2759 1.296 4.0 8 SCI-2D 4
2772 1.305 4.5 8 SCI-2D 4
2779 1.312 5.0 8 SCI-2D 4
2783 1.317 5.5 8 SCI-2D 4
2788 1.569 3.0 6 SSI-2D 4
2844 1.609 3.5 6 SSI-2D 4
2872 1.655 4.0 6 SSI-2D 4
2988 1.691 2.0 8 SSI-1D 4
3177 1.698 2.5 8 SSI-1D 4
3289 1.739 3.0 8 SSI-1D 4
3360 1790 3.5 8 SSI-1D 4
3395 1.850 4.0 8 SSI-1D 4
3551 1.868 2.5 10 SSI-1D 4
3690 1.919 3.0 10 SSI-1D 4
Family
4 ap.
SCI-1D
1 m Diam.
Family
8 ap.
SCI-2D
4 m Diam.
Family
8 ap.
SSI-1D
4 m Diam.
Intersection
of Multiple
Families
Family
4 ap.
SCI-1D
2 m Diam.
Family
6 ap.
SCI-2D
4 m Diam.
Family
6 ap.
SSI-2D
4 m Diam.
Family
10 ap.
SSI-1D
4 m Diam.
Transition from
SCI to SSI
Designs
Mission Cost
& Performance
Low
Medium
High
23 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Multi-Objective Optimization Example:
Broadband Communication Satellite Constellation
Goal: Determine with minimal computational effort a 4-
dimensional Pareto optimal set.
Broadband Design Goals: To simultaneously
– Minimize Lifecycle Cost
– Maximize Lifecycle Performance (# T1 minutes provided)
– Maximize # Satellites in View Over Market Served
– Maximize Coverage Over Populated Globe
Key Question: Is it better to find and then combine a series of 2-
dimensional P-optimal sets or attempt to simultaneously optimize
all of the metrics of interest.
Pareto Optimality: A set of design architectures in which the
systems engineer cannot improve one metric of interest without
adversely affecting at least one other metric of interest. This set
quantitatively captures the trades between the design decision
criteria.
24 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
The Broadband GINA Model
Inputs (Design Vector)
Key Outputs
Throughput
Lifecycle Cost
Market Capture
Revenues
Net Present Value
Availability
Cost Per T1 Minute
# Orbital Planes
….
Constellation Altitude
Payload &
S/C Bus
Launch &
Operations
Market
Analysis
Orbital
Dynamics
Link
Budgets
Systems
GINA
MATLAB Models
Antenna Power
1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
60
70
80
90
100
Economic
Analysis
….
Inclination
# S/C per Plane
….
Antenna Area
….
25 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Case 1 – Multi-Objective Optimization
Objective: Minimize LCC & Maximize Performance
Optimization
Formulation
& Pareto Plots
# Pareto Optimal Designs Found (60 Iterations)
126412
4-Dimensional P-Opt.LCC vs. Global
Population Coverage
LCC vs. Mean #
Satellites in View
LCC vs. Performance
10
1
10
1
Objective:
Constraints:
Isolati
() AND
on 90%
(
/ 4 4
)
.
y
y
y
y
Subje
M
ct to
MAE
Eb No
in
Max
φΓ
ΨΓ
=
=
|
|
≥
≥
5
min
db
6.0 db
Integrity 10
Rate 1.54 Mbps Per Link
Availability 10
(
Link Margin
BER
R
P
ε
?
≥
≤
≥
≥
D
) 98%Coverage ≥
26 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Case 2 – Multi-Objective Optimization
Objective: Minimize LCC & Maximize Mean # Satellites in View
# Pareto Optimal Designs Found (60 Iterations)
115118
4-Dimensional P-Opt.LCC vs. Global
Population Coverage
LCC vs. Mean #
Satellites in View
LCC vs. Performance
10
1
480
1
1
Objective:
Constraints:
() A
Isolation 90%
ND
48
0
y
y
n
ij
j
i
Sub
Min
SIV
n
M
jec
ME
a
to
A
x
t
φΓ
=
=
=
|
|
|
≥
5
min
/4.4 db
6.0 db
Integrity 10
Rate 1.54 Mbps Per Link
Availability 10
Eb No
Link Margin
BER
R
ε
?
≥
≥
≤
≥
≥
D
( ) 98%P Coverage ≥
Optimization
Formulation
& Pareto Plots
27 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Case 3 – Multi-Objective Optimization
Objective: Minimize LCC & Maximize Global Population Coverage
# Pareto Optimal Designs Found (60 Iterations)
4443
4-Dimensional P-Opt.LCC vs. Global
Population Coverage
LCC vs. Mean #
Satellites in View
LCC vs. Performance
Optimization
Formulation
& Pareto Plots
10
1
480
240
1
1
() AND
480
2
Objective:
Constraints:
Isolation 90%
40
y
y
ij
j
i
Subject to
M
Min
C
M
E
V
A
O
ax
φΓ
=
=
=
|
|
≥
|
5
min
/ 4.4 db
6.0 db
Integrity 10
Rate 1.54 Mbps Per Link
Availability 10
Eb No
Link Margin
BER
R
ε
?
≥
≥
≤
≥
≥
D
( ) 98%P Coverage ≥
28 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Case 4 – Multi-Objective Optimization
Objective: 4-Dimensional Simultaneous Optimization
# Pareto Optimal Designs Found (180 Iterations)
445916
4-Dimensional P-Opt.LCC vs. Global
Population Coverage
LCC vs. Mean #
Satellites in View
LCC vs. Performance
10
1
10
1
480
1
1
480
1
() AND
( ) AND
AND
480
Objective:
y
y
y
y
n
ij
j
i
ij
j
Min
Max
SIV
n
Max
COV
Max
φΓ
ΨΓ
=
=
=
=
=
|
|
|
|
|
240
1
Constraints:
048
240
i=
|
#
Optimization
Formulation
& Pareto Plots
29 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
(PA PB) PC
Multi-Objective Optimization Comparison
# 4-D Pareto Optimal Design Architectures Found
44P-Opt.4-D Simultaneous Optimization4
39(A U B) U CUnion of All Explored Designs3
21(PA U PB) U PCUnion of P-Opt. Sets2
1(PA PB) PC Intersection of P-Opt. Sets1
Size of Pareto Optimal
Set
Mathematical
Representation
Approach#
*Each case required the same amount of computational effort = 180 iterations.
(PA U PB) U PC
PA
PCPB
{TS}
(A U B) U C
A
CB
{TS}PA
PCPB
{TS}
30 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Observations
Combining a sequence of 2-D Pareto Optimal sets
via {Set Theory} is a viable approach for finding n-
dimensional P-optimal sets of design architectures.
However, it appears to be more computationally
efficient to formulate a single n-dimensional multi-
objective optimization problem, despite the
difficulty in visualizing the solution (can’t plot on
orthogonal axes, can plot on “radar plot.”)
31 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
N-Dimensional Problems
The same principles of Pareto Optimality hold for a trade space with
any number n dimensions (I.e. any number of decision criteria).
3 Criteria Example for Space-Based Radar
– Minimize(Lifecycle Cost) AND
– Minimize(Maximum Revisit Time) AND
– Maximize(Target Probability of Detection)
32 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Four Basic Tensions (Trade-offs) in
Product/System Development
Performance
Schedule Risk
Cost
One of the main jobs of the system designer (together with the
system architect) is to identify the principle tensions and resolve
them
Ref: Maier and Rechtin,
“The Art of Systems Architecting”, 2000
33 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Multiobjective Optimization and Isoperformance
Non-dominated
solutions occur, where
Isoperformance curves
are tangent to each other
Pareto-optimal
Curve
Tensions in Engineering
System Design can be
quantified
J1
J2
34 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Vector Optimization and Game Theory
x1
x2
J1 contours
J2 contours
Prisoner’s Dilemma
Region (cooperative)
Pareto Front
(The Contract Curve)
Nash Equilibrium
(uncooperative)
35 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
In Practice
Inefficient solutions are not candidates for optimality
In practice a “near-optimal” solution is acceptable
Solutions that satisfactorily terminate the decision
process are called “final solutions”
Multiple Criteria
Decision Making (MCDM)
Multiattribute
Decision/Utility Analysis
Multicriteria
Optimization*
- small # of alternatives
- environment of uncertainty
- resolving public policy problems
- e.g. nuclear power plant siting,
airport runway extensions ...
- large # of feasible alternatives
- deterministic environment
- less controversial problems
- business and design problems
Ref: Keeney & Raiffa, 1976
36 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Lecture Summary
Two fundamental approaches to MOO
– Scalarization of multiple objectives to a single combined
objective (e.g. Utility Theory)
– Pareto Approach with a posteriori selection
Methods for computing Pareto Front
– Weighted Sum Approach (and variants)
– Design Space Exploration + Pareto Filter
– Normal Boundary Intersection (NBI)
– Multiobjective Heuristic Algorithms
Resolving Tradeoffs are an essential part of System
Optimization
37 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
References
Edgeworth, F.Y., Mathematical Psychics, P. Keagan, London, England, 1881.
Pareto, V., Manuale di Economia Politica, Societa Editrice Libraria, Milano, Italy, 1906.
Translated into English by A.S. Schwier as Manual of Political Economy, Macmillan,
New York, 1971.
Ehrgott, M., Multicriteria Optimization, Springer-Verlag, New York, NY, 2000.
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.
Stadler, W., Multicriteria Optimization in Engineering and in the Sciences, Plenum Press,
New York, NY, 1988.
Steuer, Ralph, “Multiple Criteria Optimization - Theory, Computation and Application”,
1985
38 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Lab#3: Friday - MO in iSIGHT
iSIGHT is set
up to do
Weighted
Sum
optimization
Note
Weights and
Scale
Factors in
Parameters
Table
39 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Lab #3: Multiobjective Optimization Game
Task: Find an optimal layout for a new city, which comprises
5x5 sqm and 50’000 inhabitants that will satisfy multiple
disparate stakeholders.
5 miles
5 miles
Stakeholder groups:
a) Local Greenpeace Chapter
b) Chamber of Commerce
c) City Council (Government)
d) Resident’s Association
e) State Highway Commission
What layout should
be chosen ?
A
B
Commercial Zone (shops, restaurants, industry)
Recreational Zone (parks, lakes, forest)
Residential Zone (private homes, apartments)
1
2
3
0
Vacant Zone