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A Core Course on ModelingPowerPoint Presentation

A Core Course on Modeling

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Week 5 – Roles of Quantities in a Functional Model

- ACCEL (continued)
- a 4 categories model
- dominance and Pareto optimality
- strength algorithm

- Examples

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

2

- to-do list keeps track of incomplete expressions
- to-do list empty: script is compiled
- script compiles correctly: script starts running

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

- quantities are automatically categorized:
- x=17 constant: cat. III
- x=slider(3,0,10) user input: cat I
- x not in right hand part: output only: cat. II
- otherwise: cat. IV

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

4

- Category I:
- slider (number), checkbox (boolean), button(boolean event), input (arbitrary), cursorX, cursorY, cursorB
- cannot occur in expressions:
a=slider(10,0,20) *p

- slider with integer parameters gives integer results
- slider with 1 float parameter gives float results

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

5

- Category I:
- to use slider for non-numeric input:
r=[ch0, ch1, ch2, …, chn]

myChoice=slider(0,0,n)

p=r[myChoice]

(p can have arbitrary properties)

- to use slider for non-numeric input:

http://www.gulfdine.com/McDonald's_Markiya

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

6

- Category II:
- all cat.-II quantities are given as output
- dynamic models: p = f( p{1}, q{1} ) :
p is not in cat.-II

- to enforce a quantity in cat.-II: pp = p
- visual output with 'plot()'; this is a function and produces output cat.-II (usually 'plotOK')

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

7

- Category II:
- in IO/edit tab: show / hide values: values of all quantities
- results output: (too …) few decimals

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

8

- Category III:
- cat.-III is automatically detected for numbers or strings
- Cat-III is detected for expressions with constants only:
X = 3 * sin (7.14 / 5)

- don't use numerical constants in expressions:
x = pricePerUnit * nrUnits

x = 3.546 * nrUnits

x = 2 * PI * r (built-in constants: PI and E)

why not?

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

9

- Category IV:
- Expressionsshouldbesimple as possible:
- Prefery = x * p, p = z + t over y = x * (z+t)

- when in doubt: inspect!
- make temporary cat.-II quantity
- (even) better trick:
next week

- Expressionsshouldbesimple as possible:

image: http://shyatwow.blogspot.nl/2010/11/bug-day-inspect-bugs.html

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

10

- Category IV:
- efficiency: re-use common sub-expressions
- consider user defined functions

image: http://mewantplaynow.blogspot.nl/

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

11

- Category IV:
- efficiency: re-use common sub-expressions
- consider user defined functions

u = a + b*log(c)*sin(d)

v = e + b*log(c)*sin(d)

term = b*log(c)*sin(d)

u=a + term

v=e + term

re-using same value

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

12

- Category IV:
- efficiency: re-use common sub-expressions
- consider user defined functions

u = a + b*log(c)*sin(d)

v = e + p*log(q)*sin(r)

term(x,y,z) = x*log(y)*sin(z)

u = a + term(b,c,d)

v = e + term(p,q,r)

re-using same thinking

Week 5 – Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

13

image: http://hellnearyou.blogspot.nl/2010/06/aspria-managers-want-submission-from.html

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

14

Dominance

- Ordinal cat.-II quantities:
- C1dominates C2 C1.qi is better than C2.qi for all qi;
- ‘better’: ‘<‘ (e.g., waste) or ‘>’ (e.g., profit);
- more cat.-II quantities: fewer dominated solutions.

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

15

q2

(e.g., waste)

C3

Dominance

- Ordinal cat.-II quantities:
- C1dominates C2 C1.qi is better than C2.qi for all qi;
- ‘better’: ‘<‘ (e.g., waste) or ‘>’ (e.g., profit);
- more cat.-II quantities: fewer dominated solutions.

C2

C1 dominates C2

C1

C1 dominates C3

C2,C3: no dominance

q1(e.g., profit)

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

16

Dominance

- Only non-dominated solutions are relevant
- Dominance: prune cat.-I space;
- More cat.-II quantities: more none-dominated solutions
- nr. cat.-II quantities should be small.

image http://ornamentalplant.blogspot.nl/2011/07/trimming-pruning.html

Week 5 – Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

17

Dominance in ACCEL

- y=paretoMax(expression) enlist for maximum
- y=paretoMin(expression) enlist for minimum
- To use Pareto algorithm, express all conditions into penalties
- For inspection of the results: Paretoplot
paretoHor(x)

paretoVer(x)

Week 5 – Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

18

Dominance in ACCEL

myArea=paretoHor(paretoMax(p[myProv].area))

myPop=paretoVer(paretoMin(p[myProv].pop))

p=[Pgr,Pfr,Pdr,Pov,Pgl,Put,Pnh,Pzh,Pzl,Pnb,Pli]

myProv=slider(0,0,11)

myCap=p[myProv].cap

Pfr=['cap':'leeuwarden','pop':647239,'area':5748.74]

. . .

Pli=['cap':'maastricht','pop':1121483,'area':2209.22]

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

19

Dominance in ACCEL

- Dominated areas: bounded by iso-cat.-II quantitiy lines;
- Solutions in dominated areas: ignore;
- Non-dominated solutions: Pareto front.

D

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

20

image: http://www.usdivetravel.com/T-BolivianAndesExpedition.html

Optimization in practice

- Find 'best' concepts in cat.-I space.
- Mathematical optimization: single-valued functions.
- The 'mounteneer approach';
- Only works for 1 cat.-II quantity.

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

21

Optimization in practice

- Eckart Zitzler: Pareto + Evolution.
- genotype = blueprint of individual (‘cat.-I’);
- genotype is passed over to offspring;
- genotype phenotype, determines fitness (‘cat.-II’);
- variation in genotypes variation among phenotypes;
- fitter phenotypes beter gene-spreading.

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

22

Optimization in practice

- Start: population of random individuals (tuples of values for cat.-I quantities);
- Fitness: fitter when dominated by fewer;
- Next generation: preserve non-dominated ones;
- Complete population: mutations and crossing-over;
- Convergence: Pareto front stabilizes.

image: http://www.freakingnews.com/Mutation-Pictures---2317.asp

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

23

Optimization in practice: caveats

- Too large % non-dominated concepts: no progress;
- Find individuals in narrow niche: problematic;
- Analytical alternatives may not exist
- Need guarantee for optimal solution DON’T use Pareto-Genetic.

image: http://glup.me/epic-fail-pics-serie-196

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

24

Optimization in practice: brute force

- If anything else fails:
- local optimization for individual elements of the Pareto-front;
- Split cat.-I space in sub spaces if model function behaves different in different regimes;
- Temporarily fix some cat.-IV quantities (pretend that they are in category-III).

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

Week 5-Roles of Quantities in a Functional Model

Examples

25

paretoMax

paretoMax

paretoMin

Optimal province:

spaciousness = area / population

or

area

population

1 cat.-II quantity

2 cat.-II quantities

meaningful quantity, related to purpose

Week 5-Roles of Quantities in a Functional Model

Examples

26

paretoMin

paretoMin

paretoMin

Optimal street lamps:

efficiency = power * penalty

or

power

penalty

not too much light

not too little light

1 cat.-II quantity

2 cat.-II quantities

contrived quantity, not related to purpose

Week 5-Roles of Quantities in a Functional Model

Examples

27

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))

roadLength=40

roadWidth=15

. . .

problem: too slow to do optimization

Week 5-Roles of Quantities in a Functional Model

Examples

28

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))

roadLength=40

roadWidth=15

. . .

- Minimal intensity computed by the model

- Minimal intensity to see road marks

- Maximal intensity computed by the model

- Maximal intensity tnot to be blinded

problem: too slow to do optimization

Week 5-Roles of Quantities in a Functional Model

Examples

29

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))

roadLength=40

roadWidth=2

. . .

problem: awkward metric in cat.-II space

problem: too slow to do optimization use symmetry

Week 5-Roles of Quantities in a Functional Model

Examples

30

intPenalty

minInt

maxInt

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(log(0.00001-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP))))

roadLength=40

roadWidth=2

. . .

minP

maxP

problem: awkward metric in cat.-II space scale penalty

problem: border optima ???

Week 5-Roles of Quantities in a Functional Model

Examples

31

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,1,30)

p=slider(500.1,50,2000)

intPenalty=paretoMin(paretoHor(log(0.00001-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP))))

roadLength=40

roadWidth=2

. . .

problem: border optima ??? expand cat.-I ranges

Week 5-Roles of Quantities in a Functional Model

Examples

32

Optimal street lamps:

Summary:

- check if model exploits symmetries
- check if penalty functions represent intuition
- check if optima are not on arbitrary borders
- keep thinking: interpret trends (h 0, l0 … 1D approximation …?)

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