1. 1 A Core Course on Modeling ACCEL (continued) a 4 categories model dominance and Pareto optimality strength algorithm Examples Week 5 – Roles of Quantities in a Functional Model

2. 2 A Core Course on Modeling 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

3. 3 A Core Course on Modeling 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

4. 4 A Core Course on Modeling 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. 5 A Core Course on Modeling 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) Week 5 – Roles of Quantities in a Functional Model ACCEL: a four-categories model http://www.gulfdine.com/McDonald's_Markiya

6. 6 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')

7. 7 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

8. 8 A Core Course on Modeling 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) Week 5 – Roles of Quantities in a Functional Model ACCEL: a four-categories model why not?

9. 9 A Core Course on Modeling Week 5 – Roles of Quantities in a Functional Model ACCEL: a four-categories model image: http://shyatwow.blogspot.nl/2010/11/bug-day-inspect-bugs.html 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

10. 10 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 sub- expressions consider user defined functions image: http://mewantplaynow.blogspot.nl/

11. 11 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 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

12. 12 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 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

13. 13 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-want- submission-from.html

14. 14 A Core Course on Modeling Dominance Ordinal cat.-II quantities: C 1 dominates C 2  C 1.q i is better than C 2.q i for all q i ; ‘better’: ‘ ’ (e.g., profit); more cat.-II quantities: fewer dominated solutions. Week 5-Roles of Quantities in a Functional Model ACCEL: dominance & pareto optimality

15. 15 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model q 1 (e.g., profit) q 2 (e.g., waste) C2C2 C1C1 C3C3 C 1 dominates C 2 C 2,C 3 : no dominance C 1 dominates C 3 ACCEL: dominance & pareto optimality Dominance Ordinal cat.-II quantities: C 1 dominates C 2  C 1.q i is better than C 2.q i for all q i ; ‘better’: ‘ ’ (e.g., profit); more cat.-II quantities: fewer dominated solutions.

16. 16 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model ACCEL: dominance & pareto optimality image http://ornamentalplant.blogspot.nl/2011/07/trimming-pruning.html 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.

17. 17 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)

18. 18 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]

19. 19 A Core Course on Modeling D 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.

20. 20 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model ACCEL: strength-algorithm 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. image: http://www.usdivetravel.com/T-BolivianAndesExpedition.html

21. 21 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.

22. 22 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model ACCEL: strength-algorithm 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

23. 23 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model ACCEL: strength-algorithm 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

24. 24 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model ACCEL: strength-algorithm 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

25. 25 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model Examples Optimal province: spaciousness = area / population or area population  paretoMax  paretoMin 1 cat.-II quantity 2 cat.-II quantities meaningful quantity, related to purpose

26. 26 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model Examples Optimal street lamps: efficiency = power * penalty or power penalty  paretoMin 1 cat.-II quantity 2 cat.-II quantities not too much light not too little light contrived quantity, not related to purpose

27. 27 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

28. 28 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 Minimal intensity computed by the model Minimal intensity to see road marks Maximal intensity computed by the model Maximal intensity tnot to be blinded

29. 29 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: too slow to do optimization  use symmetry problem: awkward metric in cat.-II space

30. 30 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(log(0.00001- min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))) roadLength=40 roadWidth=2... problem: awkward metric in cat.-II space  scale penaltyproblem: border optima ??? intPenalty minPmaxP minIntmaxInt

31. 31 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(0.00001- min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))) roadLength=40 roadWidth=2... problem: border optima ???  expand cat.-I ranges

32. 32 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 …?)