![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](https://image.slidesharecdn.com/ar1twf030lecture3f-180214103216/85/Ar1-twf030-lecture3-1-Design-Optimization-14-320.jpg)
![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](https://image.slidesharecdn.com/ar1twf030lecture3f-180214103216/85/Ar1-twf030-lecture3-1-Design-Optimization-18-320.jpg)
Ar1 twf030 lecture3.1: Design Optimization
- 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.