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

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

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

4. 4 A Core Course on Modeling 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 (mm 2 ) AP=TS 2, where A is a constant (standardized: A4, A5, …) …. S = f S (T,P) or P = f P (T,S) or T = f T (P,S) ?      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model T=amount of text P=number of pages S=size of font A=area of page

5. 5 A Core Course on Modeling Elaborate each of the 3 possibilities      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model 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. 6 A Core Course on Modeling Case 1: reading light (P should be small)      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model 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=TS 2 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. 7 A Core Course on Modeling Case 2: reading easy (size of characters should be large)      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model 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. 8 A Core Course on Modeling Case 3: reading much (amount of text should be large)      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model 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 = S 2 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. 9 A Core Course on Modeling      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model quantities we need intermediate quantities quantities from context quantities we can modify Reading light: we need P; P=C/A C=TS 2 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=S 2 A  constant S  choose P  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

10. 10 A Core Course on Modeling      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model 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 S T A C P reading light P T A L R S reading easy P A S Z R T reading much general functional model (example) quantities of category II quantities of category I quantities of category III quantities of category IV P=C/A; C=TS 2 S=  L;L=R/T; R=PA T=R/Z; R=PA; Z=S 2

11. 11 A Core Course on Modeling      The 4-Categories Approach      Week 5-Roles of Quantities in a Functional Model general functional model (example) II:quantities we need I:quantities we can modify III: quantities from context IV:intermediate quantities 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.

12. 12 A Core Course on Modeling categorydepends onmeaningtypeexample Ifreely modifiablenothing modeler’s decisions, modifications, interventions, explorations … any physical dimensions, materials, brands, ‘tweakable’ parameters II express the purpose of the model I,III,IV modeler’s goals (purpose) often: ordinal profit, comfort, safety, …things for interest of the stakeholder III context (not freely modifiable) nothing beyond the authonomy of the modeler any physical constants, vendor’s catalogue, … IVauxiliary, intermediateI,III,IVinternalany(in the printer’s dilemma: Z, R, C, L) Week 5-Roles of Quantities in a Functional Model      The 4-Categories Approach     

13. 13 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model      The 4-Categories Approach      Interpretations categories I and II: purposecat.-Icat.-II predict (‘when …”)NOTHINGtime point asked for predict (‘what if …’)condition after ‘if’what is going to happen decide (e.g., design)decision quantitiesstakeholders value (profit, safety, …) steer / controlexternal perturbationdifference between desired and actual verifyNOTHINGresult of verification: true or false optimizeindependent quantityobjective quantity

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

15. 15 A Core Course on Modeling Restrictions on cat.-I quantities: The printer’s dilemma: T, S and P not all in category I, since TS 2 /P = constant. Week 5-Roles of Quantities in a Functional Model     Input of the Functional Model: Category I     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 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.

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

17. 17 A Core Course on Modeling 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. Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II    

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

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

20. 20 A Core Course on Modeling     Limitations from Context: Category III     Week 5-Roles of Quantities in a Functional Model 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. 21 A Core Course on Modeling     Intermediate Quantities: Category IV     Week 5-Roles of Quantities in a Functional Model Cat.-IV propagates values from cat.’s I,III  II cat-I cat-III cat-II cat-IV

22. 22 A Core Course on Modeling     Intermediate Quantities: Category IV     Week 5-Roles of Quantities in a Functional Model 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. 23 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     Cat. –II quantities penalize unwanted behavior of cat.-I quantities.

24. 24 A Core Course on Modeling consider the book printers’ example: three models reading light: cat.-I: S,T; cat.-II: P=TS 2 /A; q P = max(P-P 0,0) reading easy: cat.-I: T,P; cat.-II: S=  PA/T; q S = - min(S-S 0,0) reading much: cat.-I: P,S; cat.-II: T= PA/S 2 ; q T = - min(T-T 0,0) Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: 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

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

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

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

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

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

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

31. 31 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     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. 32 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     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. 33 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     Dominance: how to navigate cat.-I space

34. 34 A Core Course on Modeling 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. Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II    

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

36. 36 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     Cat.-II –space and dominance Assume cat.-II quantities ordinals; C 1 dominates C 2   q i in cat.-II, C 1.q i is better than C 2.q i ; ‘Being better’ : ‘ ’ (e.g., profit); More cat.-II quantities  fewer dominated solutions.

37. 37 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     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 Cat.-II –space and dominance

38. 38 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     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. 39 A Core Course on Modeling D. Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II     Non-dominated solutions form the Pareto – front

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

41. 41 A Core Course on Modeling 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. Week 5-Roles of Quantities in a Functional Model     Output of the Functional Model: Category II    

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

43. 43 A Core Course on Modeling     Optimality and Evolution     Week 5-Roles of Quantities in a Functional Model 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. 44 A Core Course on Modeling     Approximating the Pareto Front     Week 5-Roles of Quantities in a Functional Model 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. 45 A Core Course on Modeling     Approximating the Pareto Front     Week 5-Roles of Quantities in a Functional Model 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. 46 A Core Course on Modeling Charles Darwin     Approximating the Pareto Front     Week 5-Roles of Quantities in a Functional Model 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.

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

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

49. 49 A Core Course on Modeling     Approximating the Pareto Front     Week 5-Roles of Quantities in a Functional Model 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. 50 A Core Course on Modeling functional model helps distinguish input (choice) and output (from purpose); Building a functional model as a graph shows roles of 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 lumping and penalty functions; The distinction between requirements, desires, and wishes; The notion of dominance to 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.     Summary     Week 5-Roles of Quantities in a Functional Model