Tudelft stramien 16_9_on_optimization
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Transcript of Tudelft stramien 16_9_on_optimization
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On Evaluation & Optimization
A very short review of essential topics
Ir. Pirouz Nourian
PhD Candidate, Researcher & Instructor
@ AE+T/Chair of Design Informatics
@URBANISM/chair of Urban Design
MSc in Architecture 2009
BSc in Control Engineering 2005
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What is optimization all about?
β’ Goal-Oriented Search
β’ typically maximization or minimization
β’ Objective Function, Goal
β’ Performance Indicators
β’ Performance Optimization
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What is evaluation all about?
β’ Formulating an indicator that could
describe the performance of an
object/system according to:
β A concept of quality/fitness
β A frame of reference
β A benchmark
β An evaluation framework
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Analysis vs Evaluation
β’ Synthesis (conclusion) β Putting together various analyses
β’ Aggregation β Integral
β Sum
β Arithmetic Mean
β Harmonic Mean
β Geometric Mean
β Etcetera
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Aggregating Goals?
β’ Multi-Criteria Analysis vs Multi-Objective Optimization
β’ Weighting goals?
β’ Apples & Oranges
β’ Commensurability
β’ Dimensional Analysis
β’ WSM vs WPM in Decision Problems
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Terminology Problem Setting/Formulation
Suppose the design is formulated as a rectangle with the width W and height
H, which its area is desired to be maximized (Given the perimeter as a
constant P). In other words, the problem is to find the maximum rectangular
area that one can circumscribe with a rope of the length P. We have:
Constraint
π = 2 π+ π» = πΆπππ π‘πππ‘
Design Variable
Either W or H can be considered as a variable parameter:
πΈππ‘βππ π» =(π β 2π)
2 ππ π =
(π β 2π»)
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Objective (Fitness) Function
We can write the Area as a function of the single variable π as below:
π΄πππ π = π.π» = π.π β 2π
2= ππ/2 βπ2
Problem-Solving
π΄πππβ² π = π/2 β 2π
πΏππ‘ π΄πππβ² π =π
2β 2π = 0
π¦πππππ π = π/4 & π» = π/4
π΄ππππππ₯ = π.π» = π2/16
Solution
Perimeter Given
Maximum Area? Desired
H
W
=
= /16
W=P/4
H=P/4
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The Importance of Formulation/Design
The maximum area achieved with a rectangle is equal to W. H = π2/16, whereas if the designer in question had chosen a circle, they would have achieved the following surface area:
π΄ = ππ2, π = 2ππ = ππππ π‘.π¦πππππ π΄ = π(
π
2π)2
=π2
4π>π2
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Formulation of a Single-Objective
Optimization Problem Find a combination of the input variables that optimizes (minimizes/maximizes) a single outcome of a process:
minimizeπ₯π(π₯)
Subject to:
π π₯π β€ 0, π = 1,β¦ ,π β π₯π = 0, π = 1,β¦ , π
Where:
β’ π π₯ : βπ β β is an objective function to be minimized (or maximized) over variable π₯,
β’ π π₯π β€ 0 are constraints, and
β’ β π₯π = 0 are equality constraints.
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Find a combination of the input variables that optimizes (minimizes/maximizes) a single outcome of a process:
minimizeπ₯π(π₯)
Subject to:
π π₯π β€ 0, π = 1,β¦ ,π β π₯π = 0, π = 1,β¦ , π
Where:
β’ π π₯ : βπ β β is an objective function to be minimized (or maximized) over variable π₯,
β’ π π₯π β€ 0 are constraints, and
β’ β π₯π = 0 are equality constraints.
Formulation of a Single-Objective
Optimization Problem
Image Credit: http://www.turingfinance.com/fitness-landscape-analysis-for-computational-finance/
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Formulation of a Multi-Objective
Optimization Problem Find a combination of the input variables that optimizes (minimizes/maximizes) multiple (different, independent, and often conflicting) outcomes of a process:
minimizeπ₯[π1 π₯ , π π₯ ,β¦ , ππ(π₯)]
π . π‘. π₯ β π
Where:
β’ π: π β βπ, π π₯ = [π1 π₯ , π π₯ ,β¦ , ππ(π₯)]πis a vector-valued objective function to
be minimized over variableπ₯ β π. If an objective is to be maximized we negate it in the vector-valued objective function.
β’ Typically, there does not exist a solution optimal for all objectives; therefore we focus on Pareto-Optimal solutions; which are solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives. Technically, a solution is called Pareto Optimal if not (Pareto) dominated, that is: β A feasible solution π₯1 β π is said to dominate another solution solution π₯ β π if:
β ππ π₯1 β€ ππ π₯
for βπ β 1, π ; and βπ β 1, π such that ππ π₯1 < ππ π₯
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Formulation of a Multi-Objective
Optimization Problem Find a combination of the input variables that optimizes (minimizes/maximizes) multiple (different, independent, and often conflicting) outcomes of a process:
minimizeπ₯[π1 π₯ , π π₯ ,β¦ , ππ(π₯)]
π . π‘. π₯ β π
Where:
β’ π: π β βπ, π π₯ = [π1 π₯ , π π₯ ,β¦ , ππ(π₯)]πis a vector-valued objective function to
be minimized over variableπ₯ β π. If an objective is to be maximized we negate it in the vector-valued objective function.
β’ Typically, there does not exist a solution optimal for all objectives; therefore we focus on Pareto-Optimal solutions; which are solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives. Technically, a solution is called Pareto Optimal if not (Pareto) dominated, that is: β A feasible solution π₯1 β π is said to dominate another solution solution π₯ β π if:
β ππ π₯1 β€ ππ π₯
for βπ β 1, π ; and βπ β 1, π such that ππ π₯1 < ππ π₯
Image Credits: (Left) Enginsoft: http://www.enginsoft.com/technologies/multidisciplinary-analysis-and-optimization/multiobjective-optimization/ (Right) Professor Peter J Fleming: https://www.sheffield.ac.uk/acse/staff/peter_fleming/intromo
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Formulation of a Multi-Objective
Optimization Problem Find a combination of the input variables that optimizes (minimizes/maximizes) multiple (different,
independent, and often conflicting) outcomes of a process:
Image Courtesy of Ilya Loshchilov; http://www.loshchilov.com/publications.html
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Multiple Objectives into a Single One?
What if we want/have to find the single best solution?
Then we need to aggregate multiple objectives into one; but how?
Shall we make a weighted average of the objectives and seek to
optimize it?
Orβ¦
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Dimensional Analysis
β’ 7even Fundamental Quantities in Physics
β’ Mass, Length, Time, Electric Current,
Absolute Temperature, Amount of
Substance, Luminous Intensity
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Dimensional Analysis
β’ 7even Fundamental Quantities in Physics
From The International System of Units (SI) [8th edition, 2006; updated in 2014]
SI: By convention physical quantities are organized in a system of dimensions. Each of the seven base quantities used in the SI is regarded as having its own dimension, which is symbolically represented by a single sans serif roman capital letter. The symbols used for the base quantities, and the symbols used to denote their dimension, are given as follows.
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Dimensional Analysis
Base quantities and dimensions used in the SI
Base quantity Symbol for quantity
Symbol for dimension
SI unit
mass m M Kilogram (kg)
length l, x, r, etc. L Meter (m)
time, duration t T Second (s)
electric current I, i l Ampere (A)
absolute temperature T Ξ Kelvin (K)
amount of substance n N Mole (mol)
luminous intensity I v J Candela (cd)
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Dimensional Analysis
Base quantities and dimensions used in the SI
All other quantities are derived quantities, which may be written in terms of the base quantities by the equations of physics. The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity Q is written in the form of a dimensional product,
dimπ = ππΌπΏπ½ππΎπΌπΏΞ π π½ where the exponents πΌ, π½, πΎ, πΏ, ν, ν, and ν, which are generally small integers which can be positive, negative or zero, are called the dimensional exponents. The dimension of a derived quantity provides the same information about the relation of that quantity to the base quantities as is provided by the SI unit of the derived quantity as a product of powers of the SI base units.
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Dimensional Analysis
Example: What is the dimension of Energy?
Mechanical Energy can be the work of a force along a displacement, that is found by the dot product of the two vectors as a scalar:
π = π.π« While force can be described according to the Newtonβs Second Law, as what is needed to accelerate a mass:
π = ππ Where acceleration can be described in terms of changes in velocity of a moving object as below:
π =βπ½
βπ‘
And velocity can be formulated as the rate of displacement over time:
π½ =βπ
βπ‘
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Dimensional Analysis
Example: What is the dimension of Energy?
Mechanical Energy can be the work of a force along a displacement, that is found by the dot product of the two vectors as a scalar:
π = π.π« While force can be described according to the Newtonβs Second Law, as what is needed to accelerate a mass:
π = ππ Where acceleration can be described in terms of changes in velocity of a moving object as below:
π =βπ½
βπ‘
And velocity can be formulated as the rate of displacement over time:
π½ =βπ
βπ‘β π ππ π½ = πΏπβ1
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Dimensional Analysis
Example: What is the dimension of Energy?
Mechanical Energy can be the work of a force along a displacement, that is found by the dot product of the two vectors as a scalar:
π = π.π« While force can be described according to the Newtonβs Second Law, as what is needed to accelerate a mass:
π = ππ Where acceleration can be described in terms of changes in velocity of a moving object as below:
π =βπ½
βπ‘β π ππ π = πΏπβ
And velocity can be formulated as the rate of displacement over time:
π½ =βπ
βπ‘β π ππ π½ = πΏπβ1
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Dimensional Analysis
Example: What is the dimension of Energy?
Mechanical Energy can be the work of a force along a displacement, that is found by the dot product of the two vectors as a scalar:
π = π.π« While force can be described according to the Newtonβs Second Law, as what is needed to accelerate a mass:
π = ππ β π ππ π = ππΏπβ Where acceleration can be described in terms of changes in velocity of a moving object as below:
π =βπ½
βπ‘β π ππ π = πΏπβ
And velocity can be formulated as the rate of displacement over time:
π½ =βπ
βπ‘β π ππ π½ = πΏπβ1
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Dimensional Analysis
Example: What is the dimension of Energy?
Mechanical Energy can be the work of a force along a displacement, that is found by the dot product of the two vectors as a scalar:
π = π.π« β π ππ π = ππΏ πβ While force can be described according to the Newtonβs Second Law, as what is needed to accelerate a mass:
π = ππ β π ππ π = ππΏπβ Where acceleration can be described in terms of changes in velocity of a moving object as below:
π =βπ½
βπ‘β π ππ π = πΏπβ
And velocity can be formulated as the rate of displacement over time:
π½ =βπ
βπ‘β π ππ π½ = πΏπβ1
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Dimensional Analysis
Example: What is the dimension of Energy?
Therefore, the dimension of energy (in any form) is equal to the dimension of energy in mechanical form and equal to:
dimπΈ = ππΏ πβ
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Dimensional Analysis
Long Story Short: Apples & Oranges cannot be
compared (Added, Subtracted, Averaged)!
We can only compare (and thus add or subtract) quantities of the same dimension. It can be readily seen that we cannot get an average nor a weighted average of quantities of different physical dimensions, as that would entail adding incommensurate quantities.
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Apples & Oranges
Addition, Subtraction and Arithmetic Averages
are senseless for incommensurate quantities
We can only compare (and thus add or subtract) quantities of the same dimension. It can be readily seen that we cannot get an average nor a weighted average of quantities of different physical dimensions, as that would entail adding incommensurate quantities.
Image Credit: Paul CΓ©zanne, Still Life with Apples and Oranges
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Combining Goals/Criteria
Weighted Sum Model & Weighted Product Model
For commensurate goals/criteria:
π₯ = π€ππ₯πππ=1
π€πππ=1
or π₯ = π€ππ₯πππ=1 if weights are normalized; i.e. π€π
ππ=1 = 1
For incommensurate goals/criteria:
π₯ = π₯ππ€ππ
π=1
1
π€πππ=1 or π₯ = π₯π
π€πππ=1 if weights are normalized
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Combining Goals/Criteria
Fuzzy Aggregation
For goals that are fuzzifiable (those for which bounds or benchmarks are known): ππππβ π΄ππ·: π₯π
π
β minπ{π₯π}
ππππβ ππ : π₯ππ
β maxπ{π₯π}
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Questions?
β’ Be careful with making claims about optimized designs β’ Remember that evaluation is not equal to analysis/simulation β’ Problem Formulation is more important than problem solving β’ Optimization is not a solution to all problems in design β’ All goals cannot be dealt with at once; as there is usually a hierarchy of
issues β’ A bad design cannot be corrected with optimization β’ Optimization is merely about searching within the possibilities created
by yourself; try to give rise to good possibilities.
THANKS FOR YOUR ATTENTION