1 a core course on modeling contents functional models the 4 categories approach constructing the...

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A Core Course on Modeling

Contents

Functional Models

The 4 Categories Approach

• Constructing the Functional Model

•Input of the Functional Model: Category I

•Output of the Functional Model: Category II

•Limitations from Context: Category III

•Intermediate Quantities: Category IV

• Optimality and Evolution

• Example / Demo• Summary

• References to lecture notes + book

• References to quiz-questions and homework assignments (lecture notes)

Week 5-Roles of Quantities in a Functional Model

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Contents

Functional Model: a model with ‘inputs’ mapped to ‘outputs’

Examples:purpose predict (1 ‘when …’): input = EMPTY; output = time point

purpose predict (2 ‘what if …’): input = if-condition; output = what will happen

purpose decide: input = decision; output = consequence

purpose optimize: input = independent quantity; output = target (objective, …)

purpose steer/control: input = perturbations; output = difference between realized and desired value

purpose verify: input = EMPTY; output = succeed or fail (true or false)


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The 4-Categories Approach

the printer’s dilemma: reading lightreading light, reading easy or reading much?ing much?


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the printer’s dilemma: reading light, reading easy or reading much?

T = amount of text (char.-s)

S = size of font (mm)

P = number of pages (1)

A = area of one page (mm2)

AP=TS2,

where A is a constant (standardized: A4, A5, …)

…. but do we have S=fS(T,P) or P=fP(T,S) or T=fT(P,S) ?



T=amount of textP=number of pagesS=size of fontA=area of page

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Unclarity about ‘what depends on what’ is a main source of confusion in functional models.

the printer’s dilemma: reading light, reading easy or reading much?reading light, reading easy or reading much?




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Elaborate each of the 3 possibilities



Recollect: to go from conceptual model to formal model:

•start with quantity you need for the purpose•put this on the to-do list•while the todo list is not empty:

•take a quantity from the todo list•think: what does it depend on?

•if depends on nothing substitute constant value (perhaps with uncertainty bounds)

•else give an expression for it•if possible, use dimensional analysis•propose suitable mathematical expression•think about assumptions•in any case, verify dimensions

•add newly introduced quantities to the todo list

•todo list is empty: evaluate your model•check if purpose is satisfied; if not, refine your model


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Case 1: reading light (P should be small)



Quantity needed for purpose: P

pick P from to do list: P depends on C (=covered area), A

Expression: P=C/A

pick C from to do list: C depends on T, S

Expression: C=TS2

pick A from list constant

pick T from list choose

pick S from list choose

T=amount of textP=number of pagesS=size of fontA=area of pageC=covered area

blue, underlined quantities appear underway to express what quantities depend on

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Case 2: reading easy (size of characters should be large)



Quantity needed for purpose: S

pick S from to do list: S depends on L (= letter area = area of a single character)

Expression: S = L

pick L from to do list: L depends on R (= region covered by letters),T

Expression: L = R / T

pick R from to do list: R depends on P, A

Expression: R = P * A

pick A from list constantpick T from list choosepick P from list choose

T=amount of textP=number of pagesS=size of fontA=area of pageC=covered areaL=letter areaR=covered region

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Case 3: reading much (amount of text should be large)



Quantity needed for purpose: T

pick T from to do list: T depends on R (= region covered by letters), Z (=surface of 1 char)

Expression: T = R / Z

pick R from to do list: R depends on A, P

Expression: R = A * P

pick Z from to do list: Z depends on S

Expression: Z = S2

pick A from list constantpick S from list choosepick P from list choose

T=amount of textP=number of pagesS=size of fontA=area of pageC=covered areaL=letter areaR=covered regionZ=area 1 letter

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quantities we need

intermediate quantities

quantities from context

quantities we can modify

Reading light:

we need P; P=C/A

C=TS2

A constant

T choose

S choose

Reading easy:

we need S; S= L

L=R/T

R=PA

A constant

T choose

P choose

Reading much:

we need T; T=R/Z

R=PA

Z=S2

A constant

S choose

P chooseT=amount of textP=number of pagesS=size of fontA=area of pageC=covered areaL=letter areaR=covered regionZ=area 1 letter

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S

T

A

CP

reading light

PT

A

L

R

S

reading easy

P

A

S

Z

RT

reading much

general functional model (example)

quantities of category II

quantities of category I

quantities of category III

quantities of category IV

P=C/A; C=TS2

S=L;L=R/T; R=PA

T=R/Z; R=PA; Z=S2

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general functional model (example)

II:quantities we need

I:quantities we can modify

III: quantities from context

IV:intermediate quantities

The general Functional Model is

•a directed, a-cyclic graph

•contructed ‘from right to left’

•nodes are quantities

•arrows show dependency relations

•quantities in cat.-II: only incoming arrows

•quantities in cat.-I and cat.-III only outgoing arrows

•in cat.-IV all arrows allowed

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category depends on meaning type example

I quantity that can be freely modified

nothing modeler’s decisions, modifications, interventions,

explorations …

any physical dimensions, options, ‘tweakable’

parameters, unknowns

II quantity that expresses the need

(purpose) of the model

I,III,IV modeler’s goals (purpose)

often: ordinal

(decide, optimize,

steer/control, …)

profit, comfort, safety, …things for interest of

the stakeholder

III quantity from context (not freely modifiable)

nothing beyond the authonomy of the

modeler

any physical constants, vendor’s catalogue

data, …IV auxiliary, intermediate

quantityI,III,IV internal – only

needed to execute the model; values are ultimately irrelevant

any



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A Core Course on ModelingWeek 5-Roles of Quantities in a Functional Model


Depending on the purpose, categories I and II take different interpretations

purpose cat.-I cat.-II

predict (‘when …”) NOTHING time point asked for

predict (‘what if …’) condition after ‘if’ what is going to happen

decide (e.g., design) decision quantities stakeholders value (profit, safety, …)

steer / control external perturbation difference between desired and actual

verify NOTHING result fo verification: true or false

optimize independent quantity objective quantity

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Input of the Functional Model: Category I

The input of a functional model for design or exploration is often a complete collection of tuples.

Each of these tuple has the same properties.

Every property corresponds to one cat.-I quantity.

The input of the FM is the cartesian product of the types of all cat.-I quantities.

Example of a cat.-I space: the sandwiches of Subway with cat.-I quantities like ‘topping’, ‘addOns’, ‘typeOfBread’, ‘size’, ‘eatInOrTakeOut’, …


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Category-I quantities correspond to independent, free decisions / modifications / explorations / ….

The printer’s dilemma:

T, S and P can not all be in category I, since TS2/P=constant.

Choosing appropriate cat.-I quantities may require ‘cutting the Gordian knot’ .


Input of the Functional Model: Category I


•T,S: P may be too large to suit backpackers;

•S,P: T may be too small to suit the curious reader;

•P,T: S may be too small to suit senior readers.

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Output of the Functional Model: Category II

The model function maps decisions (=values for cat.-I quantities) into their consequences for the stakeholders.

Everything the model should yield for stakeholders, therefore is a condition on cat.-II quantities.

Designing assumes that there is something we ‘want’, and therefore some present lack of stakeholders’ value: if not, there is no need for the designed artefact.


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1. Don’t include too many cat.-II quantities;

2. Include the right cat.-II quantities;

3. Cat.-II quantities for design etc. must be ordinal;

4. Cat.-II quantities must be SMART.

remember: the design function is a model, aiming at capturing the essentials of the ATBD (there are also other reasons for a

small amount of cat.-II quantities).

Be ware of wrong optimality. E.g., when insulating your house, optimize on integral costs, not just on heating costs.

Cat.-II quantities are used to assess if one version of the ATBD is superior over another. Therefore they must allow comparison.

This includes ‘soft’ requirements (e.g., psychology, economics, …) if possible.



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Regarding SMART-ness:

Even ‘hard’ quantities (e.g., energy consumption, waste production, noise, …) often require non-trivial operationalization.

example of operationalization:

what is the energy consumption of a washing machine?

• Joule/Hour?

• Joule/wash?

• Joule/(kg wash)?

• Joule/(kg removed dirt)?

• Joule/(lifetime of the piece of laundry)?

• Joule/(lifetime of the washing machine)?



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Dilemma: many/few cat.-II quantities?

consider the book printers’ example: three models

cat.-I: S,T; cat.-II: P=TS2/A; qP= max(P-P0,0)

cat.-I: T,P; cat.-II: S=PA/T; qS= - min(S-S0,0)

cat.-I: P,S; cat.-II: T= PA/S2; qT= - min(T-T0,0)

In each model, qi expresses

something that is unwanted: the smaller qi, the better. The qi ‘punish’

unwanted behavior: penalty functions.

If nr. pages is larger than P0, qP is larger

than 0.

If point size is less than S0, qS is larger

than 0


reading light:

reading easy:

reading much:

If text is less than T0,

qT is larger than 0



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Different forms of penalties:

•y=max(x,0): it is bad if x>0

•y=|x|: it is bad if x is far from 0

•y= - min(x,0): if is bad if x<0

•y=1/|x| or 1/(+|x|), >0: it is bad if x is close to 0

(use function selector to find suitable penalty!)



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Penalty function: ‘the smaller the better’. Every qi is a cat-II quantity,

associated to a desired condition.

Adding penalty functions: Q=iqi, to

express that multiple conditions should hold simultaneously.

For Q: ‘the smaller the better’.

If separate qi non-negative, Q=0 is

ideal.

Penalty functions, like Chameleons, easily adapt to any desired condition. And they should be as small as possible, too.



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However:

adding penalty functions may violate dimension constraints;

adding penalty functions introduces (arbitrary) weights: Q=iaiqi, even if

the ai are ‘omitted’;

capitalization: express Q as a neutral quantity (e.g., € or $). With possibly non-ethical consequences.

Risks can be capitalized. But this would allow trading e.g., preventive maintenance for insurance premiums!



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Cat.-II quantities and requirements, desires, wishes

Terminology:

proposition=sentence that is true or false (‘cucumber is green’);

predicate=proposition over a concept (‘isGreen(cucumber)=true’);

requirement=predicate over some concept that needs to hold;

desire=predicate over some concept that is appreciated.



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Cat.-II quantities and requirements, desires, wishes

A third condition-type is the wish:

‘cat.-II quantity q should be as large (small) as possible’.

This, however, is impossible to achieve: it would require all possible outcomes to compare with.

Weaker version:

‘q should approximate the max (min) as achievable in the cat.-I space’.

Conditions ‘as large (small) as possible’ can not be realized: we have to restrict the search to the cat.-I space.



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Cat.-II –space and dominance

Cat.-I space contains all possible configurations of the modeled system;

This space is much too large for systematic exploration, or finding ‘good’ solutions;

‘The best’ solution will, in general not exist since various cat.-II quantities cannot be compared (e.g., different dimensions);

So: we must try to prune cat.-I space.

Cat.-I space, for all but trivial problems, is by far too large to systematically explore for man … , much like physical space.



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Cat.-II –space and dominanceAssume cat.-II quantities are ordinals:

Every axis in cat.-II space is ordered;

concept C1 dominates C2 iff, for all

cat.-II quantities qi, C1.qi is better

than C2.qi;

‘Being better’ may mean ‘<‘ (e.g., waste) or ‘>’ (e.g., profit);

If C1 dominates C2, this no longer

needs to be true if we add a further cat.-II quantity;

The more cat.-II quantities, the fewer dominated solutions.

‘Dominance’ means: being better in all respects. For design, this means: the artefact being better w.r.t. all (properties of all) stakeholders’ values.



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Cat.-II –space and dominanceAssume cat.-II quantities are ordinals:

Every axis in cat.-II space is ordered;

concept C1 dominates C2 iff, for all

cat.-II quantities qi, C1.qi is better

than C2.qi;

‘Being better’ may mean ‘<‘ (e.g., waste) or ‘>’ (e.g., profit);

If C1 dominates C2, this no longer

needs to be true if we add a further cat.-II quantity;

The more cat.-II quantities, the fewer dominated solutions.

‘Dominance’ means: being better in all respects. For design, this means: the artefact being better w.r.t. all (properties of all) stakeholders’ values.



q1 (e.g., profit)

q2

(e.g., waste) C2

C1

C3

C1 dominates C2

C2,C3: no dominance

C1 dominates C3

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Cat.-II –space and dominance

Only non-dominated solutions are relevant dominance allows pruning cat.-I space;

Since nr. non-dominated solutions is smaller with more cat.-II quantities, nr. of cat.-II quantities should be small;

For 2 cat.-II quantities, the cat.-II space can be visualized;

Dominance is defined, however, for any nr. cat.-II quantities.

Dominance is a simple criterion to prune cat.-I space. We only need to consider non-dominated solutions. The relative reduction is larger with fewer cat.-II quantities.



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Trade-offs and the Pareto front

In cat.-II space, dominated areas are half-infinite regions bounded by iso-coordinate lines/planes;

Solutions falling in one of these regions are dominated and can be ignored in cat.-I-space exploration;

Non-dominated solutions form the Pareto front. Cat.-II quantities f1 and f2 both need to be

minimal. A and B are non-dominated, C is dominated. Of A and B, none is better in absolute sense.

D

Solution D would dominate all other solutions – if it would exist.



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Trade-offs and the Pareto front

Relevance of Pareto-front:

•it bounds the achievable part of cat.-II space;

•solutions not on the Pareto front can be discarded;

•it exists for any model function, although in general it can only be approximated by a disjoint collection of solutions;

•if (part of) it is smooth, it defines two directions in cat.-II space: the direction of absolute improvement / deterioration, and the plane perpendicular to this direction which is tangent to the Pareto front and represents local trade-off relationships between cat.-II quantities.

Cat.-II quantities f1 and f2 both need to be minimal. A and B are non-dominated, C is dominated. Of A and B, none is better in absolute sense.

direction of absolute improvement

direction of absolute deteriorationtangent to the pareto-front: trade-offs



Bonus: when we apply a monotonous mapping to some or all cat.-II quantities, the collection non-dominated solutions stays the same. Example: it doesn’t matter if a penalty function is |a-b| or (a-b)2

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Limitations from Context: Category III

In order to evaluate a model function, we may need quantities, not in category I;

These are category-III quantities from the model context, not modifiable by the modeler;

Example: legislature, demography, physics, economy, vendor catalogues, human conditions, …

Challenge the demarcation between cat.-I and cat.–III for innovative design.

Example: when designing thermal house insulation, heat leakage through the windows occurs in the design function. If the window area is in cat.-I, zero-sized windows might be optimal. Else the window size is in cat.-III.


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Intermediate Quantities: Category IV

Start the construction of the model by introducing cat.-II;

Quantities that don’t depend on anything are cat.-I or cat.-III quantities;

All other quantities are cat.-IV quantities.

A visual impression of the design function. Green: cat.-I; grey: cat.-II; yellow: cat.-III; blue: cat.-IV.

Points represent quantities, not values

Arrows indicate functional dependency; notice: no cycles!

The entire network is constructed using the scheme of week 4, starting with cat.-II. When the to-do list is empty, all quantities are defined in terms of cat.-I and cat.-III quantities only.


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Optimality and Evolution

Our mission is to find ‘good’ or even ‘best’ concepts in cat.-I space.

Mathematical optimization regards single-valued functions;

Approach typically imitates a mountaineer climbing to the top of a (single-valued) mountain;

This would correspond to the situation of a single cat.-II quantity, or all cat.-II quantities lumped;

We seek something more generic.

Mathematical optimization attempts to find a local or even global extreme of a single-valued function. Most methods work by iteration, i.e., following a mountaineer on its route to the top. This approach would only apply to model functions in case of 1 cat.-II quantity.


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Approximating the Pareto Front Idea (Eckart Zitzler): combine Pareto and Evolution.

Main features of evolution:

•genotype encodes blueprint of individual (‘cat.-I’);

•genotype is passed over to offspring;

•new individual: genotype phenotype, determining its fitness (‘cat.-II’);

•variations in genotypes (mutation, cross-over) cause variation among phenotypes;

•fitter phenotypes have larger change of surviving, procreating, and passing their genotypes on to next generation.

Evolution as a principle for development may occur in biological and artificial systems alike


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Idea (Eckart Zitzler): combine Pareto and Evolution.

Issues to resolve:

•How to start population of random individuals (tuples of values for cat.-I quantities);

•How to define fitness fitter when dominated by fewer;

•Next generation preserve non-dominated ones; complete population with mutations and crossing-over;

•Convergence if Pareto front no longer moves.

Charles DarwinCharles Darwin

Pareto and Darwin: the dynamic duo of optimal design (under direction of E. Zitzler)

‘Strength’ (hence ‘SPEA’) is a property of an ATBD, derived from the cat.-II quantities, indicating how few it is dominated by, i.e. how fit it is.

Approximating the Pareto Front


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Caveats: Pareto-Genetic is not perfect

•If the fraction non-dominated concepts is too large, evolution makes no progress;

•If there broad niches, finding individuals in a narrow niche may be problematic;

•Approximations may fail to get anywhere near the theoretical best Pareto front.

(No guarantee that analytical alternatives exist)

DON’T use Pareto-Genetic if guarantee for optimal solution is required.


Nothing is perfect. There are cases when Pareto-genetic optimization does not meet its target, or when it should not be used.



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Charles DarwinCharles Darwindemo



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If anything else fails:

•Complementary approach: local optimization (to be applied on all elements of the Pareto-front separately);

•Split cat.-I space in sub spaces if model function behaves different in different regimes (e.g., too much cat.-I freedom may lead to bad evolution progress);

•Temporarily fix some cat.-IV quantities (pretend that they are in category-III).


If anything else fails, there are few brute-force methods that may help in difficult situations



http://www.square2marketing.com/Portals/112139/images/the-hulk-od-2003-resized-600.jpg

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•functional model helps distinguish input (choice) and output (from purpose);

•Building a functional model as a graph shows roles of quantities. These are:

•Cat.-I : free to choose;

• Models for (design) decision support: the notion of design space;

• Choice of cat.-I quantities: no dependency-by-anticipation;

•Cat.-II : represents the intended output;

• The advantages and disadvantages of lumping and penalty functions;

• The distinction between requirements, desires, and wishes;

• The notion of dominance to express multi-criteria comparison; Pareto front;

•Cat.-III : represents constraints from context;

•Cat.-IV : intermediate quantities;

•For optimization: use evolutionary approach;

• Approximate the Pareto front using the SPEA algorithm;

• Local search can be used for post-processing.

Summary


1 a core course on modeling contents functional models the 4 categories approach constructing the...

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