A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
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• 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
•to-do list keeps track of incomplete expressions
•to-do list empty: script is compiled
•script compiles correctly: script starts running
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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
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
•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
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
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•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
•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 'descartes()'; this is a function and
produces output  cat.-II (usually 'plotOK')
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
•Category II:
•in IO/edit tab: show / hide
values: values of all
quantities
•results output: (too …)
few decimals
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
•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)
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
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•Category IV:
•Expressions should be simple as possible:
•Prefer y = 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
•Category IV:
•efficiency: re-use
common subexpressions
•consider user
defined functions
image: http://mewantplaynow.blogspot.nl/
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
•Category IV:
•efficiency: re-use
common subexpressions
•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
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: a four-categories model
•Category IV:
•efficiency: re-use
common subexpressions
•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)
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: dominance & pareto optimality
image: http://hellnearyou.blogspot.nl/2010/06/aspria-managers-wantsubmission-from.html
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
ACCEL: dominance & pareto optimality
Dominance
• Ordinal cat.-II quantities:
• C1 dominates 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.
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
ACCEL: dominance & pareto optimality
Dominance
• Ordinal cat.-II quantities:
• C1 dominates 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.
q2
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C3
(e.g., waste)
C2
C1 dominates C2
C1 dominates C3
C1
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
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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
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)
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A Core Course on Modeling
Week 5 – Roles of Quantities in a Functional Model
ACCEL: dominance & pareto optimality
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]
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
ACCEL: dominance & pareto optimality
Dominance in ACCEL
• Dominated areas: bounded
by iso-cat.-II quantitiy lines;
• Solutions in dominated
areas: ignore;
• Non-dominated solutions:
Pareto front.
D
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
ACCEL: strength-algorithm
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image: http://www.usdivetravel.com/T-BolivianAndesExpedition.html
Optimization in practice
•Find 'best' concepts in cat.-I space.
•Mathematical optimization: singlevalued 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
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.
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
ACCEL: strength-algorithm
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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
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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
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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
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Optimal province:
paretoMax
1 cat.-II quantity
meaningful quantity,
related to purpose
spaciousness = area / population
or
area
paretoMax
population
paretoMin
2 cat.-II quantities
A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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Optimal street lamps:
paretoMin
1 cat.-II quantity
contrived
not
not too muchquantity,
light
related to purpose
not too little light
efficiency = power * penalty
or
power
paretoMin
penalty
paretoMin
2 cat.-II quantities
A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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
• Minimal
•
Minimal
roadWidth=15
intensity
• Maximal • Maximal
intensity to
computed
. . .
intensity intensity
see road
marks
by the
model
tnot to be computed
by the
blinded
model
problem: too slow to do optimization
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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
too slow to
do optimization
 use symmetry
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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intPenalty
Optimal street lamps:
minInt maxInt
dL=slider(25.5,5,50)
h=slider(5.5,3,30)
minP
p=slider(500.1,100,2000)
intPenalty=paretoMin(paretoHor(log(0.00001min(minP,minInt)+max(maxP,maxInt)-(maxP-minP))))
roadLength=40
roadWidth=2
. . .
maxP
problem: border
awkward
optima
metric
???
in cat.-II space  scale penalty
A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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(0.00001min(minP,minInt)+max(maxP,maxInt)-(maxP-minP))))
roadLength=40
roadWidth=2
. . .
problem: border optima ???  expand cat.-I ranges
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A Core Course on Modeling
Week 5-Roles of Quantities in a Functional Model
Examples
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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