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A Core Course on Modeling. Week 5 – Roles of Quantities in a Functional Model. ACCEL (continued) a 4 categories model dominance and Pareto optimality strength algorithm Examples. A Core Course on Modeling. Week 5 – Roles of Quantities in a Functional Model. ACCEL: a four-categories model.

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A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

  • ACCEL (continued)

    • a 4 categories model

    • dominance and Pareto optimality

    • strength algorithm

  • Examples


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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)

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


A Core Course on Modeling

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')


A Core Course on Modeling

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


A Core Course on Modeling

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?


A Core Course on Modeling

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

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


A Core Course on Modeling

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/


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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.


A Core Course on Modeling

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)


A Core Course on Modeling

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


A Core Course on Modeling

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)


A Core Course on Modeling

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]


A Core Course on Modeling

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


A Core Course on Modeling

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.


A Core Course on Modeling

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.


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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


A Core Course on Modeling

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 ???


A Core Course on Modeling

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


A Core Course on Modeling

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, l0 … 1D approximation …?)


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