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A Core Course on Modeling

define. Right concepts?. Right problem?. conceptualize. Right model?. formalize. Right outcome?. execute. Right answer ?. conclude. A Core Course on Modeling. Week 1- No Model Without a Purpose.     The modeling process    . 1. formulate purpose. identify entities.

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A Core Course on Modeling

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  1. define Right concepts? Right problem? conceptualize Right model? formalize Right outcome? execute Right answer? conclude A Core Course on Modeling Week 1- No Model Without a Purpose   The modeling process  1 formulate purpose identify entities choose relations obtain values formalize relations operate model obtain result present result interpret result

  2. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    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

  3. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     3 the printer’s dilemma: reading light, reading easy or reading much?

  4. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     4 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, …) T=amount of text P=number of pages S=size of font A=area of page …. S = fS(T,P) or P = fP(T,S) or T = fT(P,S) ?

  5. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     5 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 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 T=amount of text P=number of pages S=size of font A=area of page

  6. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     6 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; add C and A to list pick C from to do list: C depends on T, S (add to list) Expression: C=TS2 pick A from list  constant pick T from list  choose pick S from list  choose T=amount of text P=number of pages S=size of font A=area of page C=covered area

  7. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     7 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) Expression: S =  L; add L to list 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  constant pick T from list  choose pick P from list  choose T=amount of text P=number of pages S=size of font A=area of page C=covered area L=letter area R=covered region

  8. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     8 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; add R and Z to list 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  constant pick S from list  choose pick P from list  choose T=amount of text P=number of pages S=size of font A=area of page C=covered area L=letter area R=covered region Z=area 1 letter

  9. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     9 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  choose quantities we need intermediate quantities quantities from context quantities we can modify T=amount of text P=number of pages S=size of font A=area of page C=covered area L=letter area R=covered region Z=area 1 letter

  10. reading light S S reading much P T Z P T C R A A T P L S R reading easy A A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     10 P=C/A; C=TS2 T=R/Z; R=PA; Z=S2 general functional model (example) quantities of category I S=L;L=R/T; R=PA quantities of category II T=amount of text P=number of pages S=size of font A=area of page C=covered area L=letter area R=covered region Z=area 1 letter quantities of category IV quantities of category III

  11. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     11 Functional Model … is a directed, a-cyclic graph contructed ‘from right to left’; nodes: quantities; arrows: 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. general functional model (example) I:quantities we can modify II:quantities we need IV:intermediate quantities III: quantities from context

  12. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     12

  13. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     13 Interpretations categories I and II:

  14. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Input of the Functional Model: Category I    14 Input of functional model for design orexploration is the cartesian product of the types of all cat.-I quantities.

  15. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Input of the Functional Model: Category I    15 Restrictions on cat.-I quantities: The printer’s dilemma: T, S and P not all in category I, since TS2/P = constant. • 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. T=amount of text P=number of pages S=size of font A=area of page C=covered area L=letter area R=covered region Z=area 1 letter

  16. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    16 Everything the model should yield for stakeholders, is a condition on cat.-II quantities.

  17. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    17 Restrictions on cat.-II quantities: Don’t include too many cat.-II quantities; Include the right cat.-II quantities; Cat.-II quantities for design etc. must be ordinal; Cat.-II quantities must be SMART.

  18. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    18 Regarding SMART-ness: Low 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)?

  19. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Limitations from Context: Category III    19 Cat.-III serves to enforce plausible design solutions

  20. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Limitations from Context: Category III    20 Cat.-III quantities Evalutation of model function may need non-cat.-I quantities; Cat-III quantities: not modifiable by the modeler; Examples: legislature, demography, physics, economy, vendor catalogues, human conditions, … Innovative design challenges border between cat.-I and cat.–III.

  21. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Intermediate Quantities: Category IV    21 cat-I cat-II cat-IV cat-III Cat.-IV propagates values from cat.’s I,III  II

  22. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Intermediate Quantities: Category IV    22 Cat.-IV quantities Construction of the model starts by introducing cat.-II; Non-dependent quantities are cat.-I or cat.-III quantities; All other quantities are cat.-IV quantities.

  23. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    23 Cat. –II quantities penalize unwanted behavior of cat.-I quantities.

  24. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    24 consider the book printers’ example: three models reading light: cat.-I: S,T; cat.-II: P=TS2/A; qP = max(P-P0,0) reading easy: cat.-I: T,P; cat.-II: S=PA/T; qS= - min(S-S0,0) reading much: cat.-I: P,S; cat.-II: T= PA/S2; qT= - min(T-T0,0) T=amount of text P=number of pages S=size of font A=area of page C=covered area L=letter area R=covered region Z=area 1 letter

  25. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    25 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

  26. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    26 Cat.-II quantities and penalty functions: Every qi in cat-II, associated to a desired condition. Multiple conditions adding penalty functions: Q = iqi For Q: ‘the smaller the better’. If all qi 0, Q = 0 is ideal.

  27. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    27 Consequences of injudiciously adding penalty functions

  28. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    28 Cat.-II quantities to express conditions on output: adding qi may violate dimension constraints; introduce arbitrary weights: Q = iaiqi; capitalization: express Q in e.g. € or $ may have non-ethical consequences.

  29. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    29 Talking about requiremens, desires, wishes

  30. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    30 Cat.-II quantities and requirements, desires, wishes requirement = statement about a concept that needs to hold; desire = statement about some concept that is appreciated.

  31. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    31 Cat.-II quantities and requirements, desires, wishes requirement = statement about a concept that needs to hold; desire = statement about some concept that is appreciated; wish= q should be as large (small) as possible. Impossible: would require all outcomes to compare with. Weaker version: q should be the max (min) over cat.-I space.

  32. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    32 Cat.-II quantities and requirements, desires, wishes requirement = statement about a concept that needs to hold; desire = statement about some concept that is appreciated; wish= q should be as large (small) as possible. Impossible: would require all outcomes to compare with. Even weaker version: q should approximate the max (min) over cat.-I space.

  33. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    33 Dominance: how to navigate cat.-I space

  34. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    34 Cat.-II –space and dominance Problem 1: Cat.-I space: all possible configurations of modeled system; Much too large for systematic exploration; Problem 2: Cat.-II quantities cannot be compared  no ‘best’ solution.  Try to focus on fewer solutions.

  35. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    35 Dominance = being better in all respects

  36. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    36 Cat.-II –space and dominance Assume cat.-II quantities ordinals; C1dominates C2qi in cat.-II, C1.qi is better than C2.qi; ‘Being better’ : ‘<‘ (e.g., waste) or ‘>’ (e.g., profit); More cat.-II quantities  fewer dominated solutions.

  37. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    37 Cat.-II –space and dominance C3 q2 (e.g., waste) C2 C1 dominates C2 C1 C1 dominates C3 C2,C3: no dominance q1(e.g., profit)

  38. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    38 Cat.-II –space and dominance Relevant solutions are non-dominated  dominated solutions are irrelevant  it is allowed to consider fewer solutions; # non-dominated solutions decreases with increasing # cat.-II quantities  # cat.-II quantities should be small.

  39. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    39 D . Non-dominated solutions form the Pareto – front

  40. direction of absolute deterioration tangent to the pareto-front: trade-offs direction of absolute improvement A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    40

  41. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    41 Trade-offs and the Pareto front Pareto-front bounds achievable part of cat.-II space; Solutions not on the Pareto front can be discarded; Exists for any model function; It is approximated by a disjoint collection of solutions.

  42. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Optimality and Evolution    42 Limitations to mathematical optimization

  43. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Optimality and Evolution    43 Optimality Find ‘good’ or even ‘best’ concepts in cat.-I space. Mathematical optimization: single-valued functions; Approach: mountaineer climbing to a top of a mountain; Corresponds to single cat.-II quantity, or full lumping; What to do for multiple cat.-II quantities?

  44. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    44 Eckart Zitzler: Pareto  Evolution genotype encodes blueprint of individual (‘cat.-I’); genotype passed over to offspring; new individual: genotype  phenotype; phenotype determining fitness (‘cat.-II’); variations in genotypes  variation among phenotypes; fitter phenotypes: larger change of surviving, procreating, and passing their genotypes on to next generation.

  45. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    45 Eckart Zitzler: Pareto  Evolution Start  population of random individuals; Fitness  fitter when dominated by fewer; Next generation  preserve non-dominated ones; Complete population with mutations and crossing-over; Convergence  Pareto front no longer moves.

  46. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    46 Eckart Zitzler: Pareto  Evolution Too large % non-dominated concepts: no progress; Broad niches: difficult to find good individuals in a narrow niche; Approximations: don’t get near theoretically best Pareto front; No guarantee that analytical alternatives exist. DON’T use Pareto-Genetic if optimal solution is required. Charles Darwin

  47. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    47 demo

  48. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    48 http://www.square2marketing.com/Portals/112139/images/the-hulk-od-2003-resized-600.jpg Brute force if anything else fails

  49. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    49 If anything else fails: Local optimization for all elements of the Pareto-front separately; Split cat.-I space in sub spacesfor different regimes; Temporarily fix some cat.-IV quantities.

  50. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Summary    50 • functional modelhelps distinguish input (choice) and output (from purpose); • Building a functional model as a graph shows rolesof 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 lumpingand penalty functions; • The distinction between requirements, desires, and wishes; • The notion of dominanceto 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.

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