Lecture notes of the course Future Models I (AR1TWF030), The Why Factory, Directed by Prof. Winy Mass, TU Delft, Faculty of Architecture and Built Environment

1. 11
On Design Optimization:
Preliminaries of Design Performance Optimization
Dr.ir. Pirouz Nourian
Assistant Professor of Design Informatics
Department of Architectural Engineering & Technology
Faculty of Architecture and Built Environment

2. 22
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability
Performance: Measurable Functionality of a Designed Artefact
https://commons.wikimedia.org/wiki/File:Zencars_(Tazzari_Zero)_at_Avenue_Louise,_Brussels,_Belgium.jpghttps://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Webber_usgp_2004.jpg/1280px-Webber_usgp_2004.jpg

3. 33
https://www.designingbuildings.co.uk/wiki/Passive_building_design
Design Principles are more important than Design Optimization
Performance
Design Principles
What is Optimization
Forward vs Backward
Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

4. 44
What is optimization all about?
• Performance: Measurable Functionality
• Performance Optimization
• Performance Indicators
• Objective Function, Goal
• Typically Maximization or Minimization
• Mathematical Problem Solving (Feedforward)
• Goal-Oriented Search (Feedback)
Performance
Design Principles
What is Optimization
Forward vs Backward
Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

5. 55
What is optimization all about?
• Mathematical Problem Solving (Feedforward)
• E.g.
• Goal-Oriented Search (Feedback)
• E.g.
Parametric
Circle
Radius𝑟 = ൗ𝐴
𝜋
A 100 𝑚2
big circle
Parametric
Circle
Radius circle
Manipulate R
to minimize Δ
Compute
Area
How do we make a circle with the area of 100 𝑚2
?
How do we make a circle with the area of 100 𝑚2
?
Performance
Design Principles
What is Optimization
Forward vs Backward
Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

6. 66
What is evaluation all about?
Formulating an indicator that could describe the
performance of an object/system according to:
– A concept of quality/fitness
– A benchmark (such as minimum and/or maximum values)
– A frame of reference (e.g. daylight guidelines & regulations)
– An evaluation framework (e.g. LEED or BREAM)
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

7. 77
Spatially and/or Temporally Complex Performance:
Analysis/Simulation vs Evaluation
• Synthesis (conclusion)
– Putting together various analyses
• Aggregation
– Integral
– Sum
– Arithmetic Mean
– Harmonic Mean
– Geometric Mean
– Etcetera
• Comparison
– Normalization/Relativization against benchmarks
– Mapping relative quality in reference to an evaluation framework
http://www.formfollowsperformance.com/tag/daylight-simulation/page/2/
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

8. 88
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 t
he length P. We have:
Constraint
𝑃 = 2 𝑊 + 𝐻 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Design Variable
Either W or H can be considered as a variable parameter:
𝐸𝑖𝑡ℎ𝑒𝑟 𝐻 =
(𝑃 − 2𝑊)
2
𝑜𝑟 𝑊 =
(𝑃 − 2𝐻)
2
Objective (Fitness) Function
We can write the Area as a function of the single variable 𝑊 as below:
𝐴𝑟𝑒𝑎 𝑊 = 𝑊. 𝐻 = 𝑊.
𝑃 − 2𝑊
2
= 𝑃𝑊/2 − 𝑊2
Problem-Solving
𝐴𝑟𝑒𝑎′
𝑊 = 𝑃/2 − 2𝑊
𝐿𝑒𝑡 𝐴𝑟𝑒𝑎′
𝑊 =
𝑃
2
− 2𝑊 = 0
𝑦𝑖𝑒𝑙𝑑𝑠
𝑊 = 𝑃/4 & 𝐻 = 𝑃/4
𝐴𝑟𝑒𝑎 𝑚𝑎𝑥 = 𝑊. 𝐻 = 𝑃2
/16
Solution
Perimeter  Given
Maximum Area?  Desired
H
W
=
= 2
/16
W=P/4
H=P/4
Single Objective, Simple Performance
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

9. 99
The Importance of Formulation/Design
The maximum area achieved with a rectangle is equal to W. H = 𝑃2/16, whereas if the designer in
question had chosen a circle, they would have achieved the following surface area:
𝐴 = 𝜋𝑟2, 𝑃 = 2𝜋𝑟 = 𝑐𝑜𝑛𝑠𝑡.
𝑦𝑖𝑒𝑙𝑑𝑠
𝐴 = 𝜋(
𝑃
2𝜋
)2=
𝑃2
4𝜋
>
𝑃2
16
• If something is not on the internet it cannot be found even by Google!
• Design principles are far more important than any optimization process.
• A bad design cannot be corrected by any optimization process.
• Optimization in an absolute sense is irrelevant for design products, because:
• Any design can be optimized within the boundaries defined by its primary formulation.
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

10. 1010
Formulation of a Single-Objective Optimization Problem
Find a combination of the input variables that optimizes (minimizes/maximizes) a single outcome
of a process:
Image Credit: http://www.turingfinance.com/fitness-landscape-analysis-for-computational-finance/
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

11. 1111
Formulation of a Single-Objective Optimization Problem
Find a combination of the input variables that optimizes (minimizes/maximizes) a single outcome
of a process:
maximize
𝑥
𝑓(𝑥)
Subject to:
𝑔𝑖 𝑥 ≤ 0, 𝑖 = 1,2, … , 𝑚
ℎ𝑗 𝑥 = 0, 𝑗 = 1,2, … , 𝑝
Where:
• 𝑓 𝑥 : ℝ 𝑛
→ ℝ is an objective function to be minimized (or maximized) over variable 𝑥,
• 𝑔𝑖 𝑥 ≤ 0 are constraints, and
• ℎ𝑗 𝑥 = 0 are equality constraints.
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

12. 1212
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:
𝑓𝑖 𝑥1
≤ 𝑓𝑖 𝑥2
for ∀𝑖 ∈ 1, 𝑘 ; and ∃𝑗 ∈ 1, 𝑘 such that 𝑓𝑗 𝑥1
< 𝑓𝑗 𝑥2
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
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

13. 1313
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:
𝑓𝑖 𝑥1
≤ 𝑓𝑖 𝑥2
for ∀𝑖 ∈ 1, 𝑘 ; and ∃𝑗 ∈ 1, 𝑘 such that 𝑓𝑗 𝑥1
< 𝑓𝑗 𝑥2
Image Courtesy of Ilya Loshchilov; http://www.loshchilov.com/publications.html
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

14. 1414
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:
minimize
𝑥
[𝑓1 𝑥 , 𝑓2 𝑥 , … , 𝑓𝑘(𝑥)]
𝑠. 𝑡. 𝑥 ∈ 𝑋
Where:
• 𝑓: 𝑋 → ℝ 𝑘
, 𝑓 𝑥 = [𝑓1 𝑥 , 𝑓2 𝑥 , … , 𝑓𝑘(𝑥)] 𝑇
is a vector-valued objective function to be minimized
over variable𝑥 ∈ 𝑋. If an objective is to be maximized we negate it in the vector-valued
objective function.
• 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:
– A feasible solution 𝑥1
∈ 𝑋 is said to dominate another solution solution 𝑥2
∈ 𝑋 if:
– 𝑓𝑖 𝑥1
≤ 𝑓𝑖 𝑥2
for ∀𝑖 ∈ 1, 𝑘 ; and ∃𝑗 ∈ 1, 𝑘 such that 𝑓𝑗 𝑥1
< 𝑓𝑗 𝑥2
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

15. 1515
Aggregating Goals?
• Multi-Criteria Analysis vs Multi-Objective
Optimization
• Weighting goals?
• Apples & Oranges
• Commensurability
• Dimensional Analysis
• WSM vs WPM in Decision Problems
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

16. 1616
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?
Or…
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

17. 1717
Dimensional Analysis
• 7even Fundamental Quantities in Physics
• Mass, Length, Time, Electric Current,
Absolute Temperature, Amount of
Substance, Luminous Intensity
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

18. 1818
Dimensional Analysis
• 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.
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

19. 1919
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 Θ Kelvin (K)
amount of substance n N Mole (mol)
luminous intensity I v J Candela (cd)
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

20. 2020
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,
dim 𝑄 = 𝑀 𝛼 𝐿 𝛽 𝑇 𝛾 𝐼 𝛿Θ 𝜀 𝑁 𝜁 𝐽 𝜂
where the exponents 𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁, and 𝜂, 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.
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

21. 2121
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:
𝑊 = 𝑭. 𝑫
While force can be described according to the Newton’s Second Law,
as what is needed to accelerate a mass:
𝑭 = 𝑚𝒂
Where acceleration can be described in terms of changes in velocity of
a moving object as below:
𝒂 =
∆𝑽
∆𝑡
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

22. 2222
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:
𝑊 = 𝑭. 𝑫
While force can be described according to the Newton’s Second Law,
as what is needed to accelerate a mass:
𝑭 = 𝑚𝒂
Where acceleration can be described in terms of changes in velocity of
a moving object as below:
𝒂 =
∆𝑽
∆𝑡
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

23. 2323
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:
𝑊 = 𝑭. 𝑫
While force can be described according to the Newton’s Second Law,
as what is needed to accelerate a mass:
𝑭 = 𝑚𝒂
Where acceleration can be described in terms of changes in velocity of
a moving object as below:
𝒂 =
∆𝑽
∆𝑡
⇒ 𝒅𝒊𝒎 𝑎 = 𝐿𝑇−2
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

24. 2424
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:
𝑊 = 𝑭. 𝑫
While force can be described according to the Newton’s Second Law,
as what is needed to accelerate a mass:
𝑭 = 𝑚𝒂 ⇒ 𝒅𝒊𝒎 𝑭 = 𝑀𝐿𝑇−2
Where acceleration can be described in terms of changes in velocity of
a moving object as below:
𝒂 =
∆𝑽
∆𝑡
⇒ 𝒅𝒊𝒎 𝑎 = 𝐿𝑇−2
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

25. 2525
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:
𝑊 = 𝑭. 𝑫 ⇒ 𝒅𝒊𝒎 𝑊 = 𝑀𝐿2 𝑇−2
While force can be described according to the Newton’s Second Law,
as what is needed to accelerate a mass:
𝑭 = 𝑚𝒂 ⇒ 𝒅𝒊𝒎 𝑭 = 𝑀𝐿𝑇−2
Where acceleration can be described in terms of changes in velocity of
a moving object as below:
𝒂 =
∆𝑽
∆𝑡
⇒ 𝒅𝒊𝒎 𝑎 = 𝐿𝑇−2
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

26. 2626
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:
dim 𝐸 = 𝑀𝐿2 𝑇−2
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

27. 2727
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.
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

28. 2828
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 Cézanne, Still Life with Apples and Oranges
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

29. 2929
Combining Goals/Criteria
Weighted Sum Model & Weighted Product Model
For commensurate goals/criteria:
ഥ𝑥 =
σ 𝑖=1
𝑛
𝑤 𝑖 𝑥 𝑖
σ𝑖=1
𝑛 𝑤𝑖
or ഥ𝑥 = σ𝑖=1
𝑛
𝑤𝑖 𝑥𝑖 if weights are normalized; i.e. σ𝑖=1
𝑛
𝑤𝑖 = 1
For incommensurate goals/criteria:
ഥ𝑥 = ς𝑖=1
𝑛
𝑥𝑖
𝑤 𝑖
1
σ 𝑖=1
𝑛 𝑤 𝑖 or ഥ𝑥 = ς𝑖=1
𝑛
𝑥𝑖
𝑤 𝑖if weights are normalized
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

30. 3030
Combining Goals/Criteria
Fuzzy Aggregation
For goals that are fuzzifiable (those for which bounds or benchmarks
are known):
𝑍𝑎𝑑𝑒ℎ 𝐴𝑁𝐷: ሩ
𝑖
𝑥𝑖 ≔ min
𝑖
{𝑥𝑖}
𝑍𝑎𝑑𝑒ℎ 𝑂𝑅: ራ
𝑖
𝑥𝑖 ≔ max
𝑖
{𝑥𝑖}
Performance
Design Principles
What is Optimization
Forward vs Backward
What is Evaluation
Terminology
Single Objective
Multiple Objectives
Dimensionality
Commensurability

31. 3131
Notes
• Be careful with making claims about optimized designs
• Remember that evaluation is not equal to analysis/simulation
• Problem Formulation is more important than problem solving
• Optimization is not a solution to all problems in design
• All goals cannot be dealt with at once; as there is usually a hierarchy of issues
• A bad design cannot be corrected with optimization
• Optimization is merely about searching within the possibilities created by yourself; try to
give rise to good possibilities.

32. 3232
Questions:
p.nourian@tudelft.nl