This is a short presentation given in the context of a computational design course for MSc architectural engineering students. It is hopefully insightful for other engineering students as well.
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
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
A machine learning method for efficient design optimization in nano-optics JCMwave
The slideshow contains a brief explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradient-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
A machine learning method for efficient design optimization in nano-opticsJCMwave
Explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradiant-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
Paper Study: Melding the data decision pipelineChenYiHuang5
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
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
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
A machine learning method for efficient design optimization in nano-optics JCMwave
The slideshow contains a brief explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradient-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
A machine learning method for efficient design optimization in nano-opticsJCMwave
Explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradiant-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
Paper Study: Melding the data decision pipelineChenYiHuang5
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
Duality Theory in Multi Objective Linear Programming Problemstheijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Study on Evaluation of Venture Capital Based onInteractive Projection Algorithminventionjournals
International Journal of Business and Management Invention (IJBMI) is an international journal intended for professionals and researchers in all fields of Business and Management. IJBMI publishes research articles and reviews within the whole field Business and Management, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
How to optimize with the help of the Particle Swarm Optimization Technique and xlOptimizer ? This brief tutorial will enable you to solve any optimization problem with the application of Particle Swarm Optimization Method. After a brief introduction about the method the tutorial will show you the steps that you will need to follow for application of PSO in optimization even if you do not know any programming.(Some basic knowledge of MS Excel 2010 and later is required).
Kyeong Soo Kim, "Atomic scheduling of appliance energy consumption in residential smart grid," Invited talk, CNU International Workshop on Industrial Mathematics, Chungnam National University (CNU), Daejeon, Korea, Oct. 7, 2016.
"Computational Support for Functionality Selection in Interaction Design" CHI...Aalto University
Talk by Andreas Karrenbauer / Max Planck Institute for Informatics. Presented at the CHI conference (chi2018.acm.org) by Andreas Karrenbauer / Max Planck. In collaboration with Anna Maria Feit, Antti Oulasvita, and Perttu Lähteenlahti
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
Duality Theory in Multi Objective Linear Programming Problemstheijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Study on Evaluation of Venture Capital Based onInteractive Projection Algorithminventionjournals
International Journal of Business and Management Invention (IJBMI) is an international journal intended for professionals and researchers in all fields of Business and Management. IJBMI publishes research articles and reviews within the whole field Business and Management, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
How to optimize with the help of the Particle Swarm Optimization Technique and xlOptimizer ? This brief tutorial will enable you to solve any optimization problem with the application of Particle Swarm Optimization Method. After a brief introduction about the method the tutorial will show you the steps that you will need to follow for application of PSO in optimization even if you do not know any programming.(Some basic knowledge of MS Excel 2010 and later is required).
Kyeong Soo Kim, "Atomic scheduling of appliance energy consumption in residential smart grid," Invited talk, CNU International Workshop on Industrial Mathematics, Chungnam National University (CNU), Daejeon, Korea, Oct. 7, 2016.
"Computational Support for Functionality Selection in Interaction Design" CHI...Aalto University
Talk by Andreas Karrenbauer / Max Planck Institute for Informatics. Presented at the CHI conference (chi2018.acm.org) by Andreas Karrenbauer / Max Planck. In collaboration with Anna Maria Feit, Antti Oulasvita, and Perttu Lähteenlahti
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
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
Ar1 twf030 lecture2.1: Geometry and Topology in Computational DesignPirouz Nourian
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
Preliminaries of Analytic Geometry and Linear Algebra 3D modellingPirouz Nourian
from my lecture notes for the course Geo1004 (2015), 3D modelling of the built environment, at TU Delft, faculty of Architecture and the Built Environment
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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Runway Orientation Based on the Wind Rose Diagram.pptx
Tudelft stramien 16_9_on_optimization
1. 1
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
2. 2
What is optimization all about?
• Goal-Oriented Search
• typically maximization or minimization
• Objective Function, Goal
• Performance Indicators
• Performance Optimization
3. 3
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 frame of reference
– A benchmark
– An evaluation framework
4. 4
Analysis vs Evaluation
• Synthesis (conclusion)
– Putting together various analyses
• Aggregation
– Integral
– Sum
– Arithmetic Mean
– Harmonic Mean
– Geometric Mean
– Etcetera
5. 5
Aggregating Goals?
• Multi-Criteria Analysis vs Multi-Objective
Optimization
• Weighting goals?
• Apples & Oranges
• Commensurability
• Dimensional Analysis
• WSM vs WPM in Decision Problems
6. 6
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
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
=
= /16
W=P/4
H=P/4
7. 7
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
8. 8
Formulation of a Single-Objective
Optimization Problem
Find a combination of the input variables that optimizes
(minimizes/maximizes) a single outcome of a process:
minimize
𝑥
𝑓(𝑥)
Subject to:
𝑔 𝑥𝑖 ≤ 0, 𝑖 = 1, … , 𝑚
ℎ 𝑥𝑖 = 0, 𝑖 = 1, … , 𝑝
Where:
• 𝑓 𝑥 : ℝ 𝑛
→ ℝ is an objective function to be minimized (or
maximized) over variable 𝑥,
• 𝑔 𝑥𝑖 ≤ 0 are constraints, and
• ℎ 𝑥𝑖 = 0 are equality constraints.
9. 9
Find a combination of the input variables that optimizes
(minimizes/maximizes) a single outcome of a process:
minimize
𝑥
𝑓(𝑥)
Subject to:
𝑔 𝑥𝑖 ≤ 0, 𝑖 = 1, … , 𝑚
ℎ 𝑥𝑖 = 0, 𝑖 = 1, … , 𝑝
Where:
• 𝑓 𝑥 : ℝ 𝑛
→ ℝ is an objective function to be minimized (or
maximized) over variable 𝑥,
• 𝑔 𝑥𝑖 ≤ 0 are constraints, and
• ℎ 𝑥𝑖 = 0 are equality constraints.
Formulation of a Single-Objective
Optimization Problem
Image Credit: http://www.turingfinance.com/fitness-landscape-analysis-for-computational-finance/
10. 10
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 , 𝑓 𝑥 , … , 𝑓𝑘(𝑥)]
𝑠. 𝑡. 𝑥 ∈ 𝑋
Where:
• 𝑓: 𝑋 → ℝ 𝑘
, 𝑓 𝑥 = [𝑓1 𝑥 , 𝑓 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 𝑥 ∈ 𝑋 if:
– 𝑓𝑖 𝑥1
≤ 𝑓𝑖 𝑥 for ∀𝑖 ∈ 1, 𝑘 ; and ∃𝑗 ∈ 1, 𝑘 such that 𝑓𝑗 𝑥1
< 𝑓𝑗 𝑥
11. 11
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 , 𝑓 𝑥 , … , 𝑓𝑘(𝑥)]
𝑠. 𝑡. 𝑥 ∈ 𝑋
Where:
• 𝑓: 𝑋 → ℝ 𝑘
, 𝑓 𝑥 = [𝑓1 𝑥 , 𝑓 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 𝑥 ∈ 𝑋 if:
– 𝑓𝑖 𝑥1
≤ 𝑓𝑖 𝑥 for ∀𝑖 ∈ 1, 𝑘 ; and ∃𝑗 ∈ 1, 𝑘 such that 𝑓𝑗 𝑥1
< 𝑓𝑗 𝑥
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
12. 12
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
13. 13
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…
14. 14
Dimensional Analysis
• 7even Fundamental Quantities in Physics
• Mass, Length, Time, Electric Current,
Absolute Temperature, Amount of
Substance, Luminous Intensity
15. 15
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.
16. 16
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)
17. 17
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.
18. 18
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
19. 19
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
20. 20
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−
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
21. 21
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−
Where acceleration can be described in terms of changes in velocity of a
moving object as below:
𝒂 =
∆𝑽
∆𝑡
⇒ 𝒅𝒊𝒎 𝑎 = 𝐿𝑇−
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
22. 22
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 = 𝑀𝐿 𝑇−
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−
Where acceleration can be described in terms of changes in velocity of a
moving object as below:
𝒂 =
∆𝑽
∆𝑡
⇒ 𝒅𝒊𝒎 𝑎 = 𝐿𝑇−
And velocity can be formulated as the rate of displacement over time:
𝑽 =
∆𝒙
∆𝑡
⇒ 𝒅𝒊𝒎 𝑽 = 𝐿𝑇−1
23. 23
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 = 𝑀𝐿 𝑇−
24. 24
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.
25. 25
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
26. 26
Combining Goals/Criteria
Weighted Sum Model & Weighted Product Model
For commensurate goals/criteria:
𝑥 =
𝑤 𝑖 𝑥 𝑖
𝑛
𝑖=1
𝑤 𝑖
𝑛
𝑖=1
or 𝑥 = 𝑤𝑖 𝑥𝑖
𝑛
𝑖=1 if weights are normalized; i.e. 𝑤𝑖
𝑛
𝑖=1 = 1
For incommensurate goals/criteria:
𝑥 = 𝑥𝑖
𝑤 𝑖𝑛
𝑖=1
1
𝑤 𝑖
𝑛
𝑖=1 or 𝑥 = 𝑥𝑖
𝑤 𝑖𝑛
𝑖=1 if weights are normalized
28. 28
Questions?
• 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.
THANKS FOR YOUR ATTENTION