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A Core Course on Modeling

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

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A Core Course on Modeling

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  1. 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 • Summary • References to lecture notes + book • References to quiz-questions and homework assignments (lecture notes)

  2. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Contents     2 Functional Model: a model with ‘inputs’ mapped to ‘outputs’ Examples: purpose predict (1 ‘when …’): input = EMPTY; output = time point purpose predict (2 ‘what if …’): input = if-condition; output = what will happen purpose decide: input = decision; output = consequence purpose optimize: input = independent quantity; output = target (objective, …) purpose steer/control: input = perturbations; output = difference between realized and desired value purpose verify: input = EMPTY; output = succeed or fail (true or false)

  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 …. but do we have 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 the printer’s dilemma: reading light, reading easy or reading much? T=amount of text P=number of pages S=size of font A=area of page Unclarity about ‘what depends on what’ is a main source of confusion in functional models.

  6. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     6 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 (perhaps 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

  7. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     7 Case 1: reading light (P should be small) blue, underlined quantities appear underway to express what quantities depend on Quantity needed for purpose: P pick P from to do list: P depends on C (=covered area), A Expression: P=C/A pick C from to do list: C depends on T, S 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

  8. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     8 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 = area of a single character) Expression: S =  L 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

  9. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     9 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 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

  10. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     10 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

  11. 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     11 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

  12. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     12 The general Functional Model is • a directed, a-cyclic graph • contructed ‘from right to left’ • nodes are quantities • arrows show 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

  13. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     13

  14. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    The 4-Categories Approach     14 Depending on the purpose, categories I and II take different interpretations

  15. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Input of the Functional Model: Category I    15 The input of a functional model for design or exploration is often a complete collection of tuples. Each of these tuple has the same properties. Every property corresponds to one cat.-I quantity. The input of the FM is the cartesian product of the types of all cat.-I quantities. Example of a cat.-I space: the sandwiches of Subway with cat.-I quantities like ‘topping’, ‘addOns’, ‘typeOfBread’, ‘size’, ‘eatInOrTakeOut’, …

  16. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Input of the Functional Model: Category I    16 Category-I quantities correspond to independent, free decisions / modifications / explorations / …. The printer’s dilemma: T, S and P can not all be 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. Choosing appropriate cat.-I quantities may require ‘cutting the Gordian knot’ . 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

  17. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    17 The model function maps decisions (=values for cat.-I quantities) into their consequences for the stakeholders. Everything the model should yield for stakeholders, therefore is a condition on cat.-II quantities. Designing assumes that there is something we ‘want’, and therefore some present lack of stakeholders’ value: if not, there is no need for the designed artefact.

  18. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    18 • 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. remember: the design function is a model, aiming at capturing the essentials of the ATBD (there are also other reasons for a small amount of cat.-II quantities). Be ware of wrong optimality. E.g., when insulating your house, optimize on integral costs, not just on heating costs. Cat.-II quantities are used to assess if one version of the ATBD is superior over another. Therefore they must allow comparison. This includes ‘soft’ requirements (e.g., psychology, economics, …) if possible.

  19. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    19 Regarding SMART-ness: Even ‘hard’ quantities (e.g., energy consumption, waste production, noise, …) often require non-trivial operationalization. example of operationalization: what is the 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)?

  20. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    20 Dilemma: many/few cat.-II quantities? consider the book printers’ example: three models cat.-I: S,T; cat.-II: P=TS2/A; qP=max(P-P0,0) cat.-I: T,P; cat.-II: S=PA/T; qS= - min(S-S0,0) cat.-I: P,S; cat.-II: T= PA/S2; qT= -min(T-T0,0) If nr. pages is larger than P0, qP is larger than 0. reading light: If point size is less than S0, qS is larger than 0 If text is less than T0, qT is larger than 0 reading easy: reading much: In each model, qi expresses something that is unwanted: the smaller qi, the better. The qi ‘punish’ unwanted behavior: penalty functions. 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

  21. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    21 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 (use function selector to find suitable penalty!)

  22. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    22 Dilemma: many/few cat.-II quantities? Penalty function: ‘the smaller the better’. Every qi is a cat-II quantity, associated to a desired condition. Adding penalty functions: Q=iqi, to express that multiple conditions should hold simultaneously. For Q: ‘the smaller the better’. If separate qi non-negative, Q=0 is ideal. Penalty functions, like Chameleons, easily adapt to any desired condition. And they should be as small as possible, too.

  23. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    23 Dilemma: many/few cat.-II quantities? However: adding penalty functions may violate dimension constraints; adding penalty functions introduces (arbitrary) weights: Q=iaiqi, even if the ai are ‘omitted’; capitalization: express Q as a neutral quantity (e.g., € or $). With possibly non-ethical consequences. Risks can be capitalized. But this would allow trading e.g., preventive maintenance for insurance premiums!

  24. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    24 Cat.-II quantities and requirements, desires, wishes Terminology: proposition=sentence that is true or false (‘cucumber is green’); predicate=proposition over a concept (‘isGreen(cucumber)=true’); requirement=predicate over some concept that needs to hold; desire=predicate over some concept that is appreciated.

  25. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    25 Cat.-II quantities and requirements, desires, wishes A third condition-type is the wish: ‘cat.-II quantity q should be as large (small) as possible’. This, however, is impossible to achieve: it would require all possible outcomes to compare with. Weaker version: ‘q should approximate the max (min) as achievable in the cat.-I space’. Conditions ‘as large (small) as possible’ can not be realized: we have to restrict the search to the cat.-I space.

  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 –space and dominance Cat.-I space contains all possible configurations of the modeled system; This space is much too large for systematic exploration, or finding ‘good’ solutions; ‘The best’ solution will, in general not exist since various cat.-II quantities cannot be compared (e.g., different dimensions); So: we must try to prune cat.-I space. Cat.-I space, for all but trivial problems, is by far too large to systematically explore for man … , much like physical space.

  27. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    27 Cat.-II –space and dominance Assume cat.-II quantities are ordinals: Every axis in cat.-II space is ordered; concept C1dominates C2 iff, for all cat.-II quantities qi, C1.qi is better than C2.qi; ‘Being better’ may mean ‘<‘ (e.g., waste) or ‘>’ (e.g., profit); If C1 dominates C2, this no longer needs to be true if we add a further cat.-II quantity; The more cat.-II quantities, the fewer dominated solutions. ‘Dominance’ means: being better in all respects. For design, this means: the artefact being better w.r.t. all (properties of all) stakeholders’ values.

  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 –space and dominance C3 Assume cat.-II quantities are ordinals: Every axis in cat.-II space is ordered; concept C1dominates C2 iff, for all cat.-II quantities qi, C1.qi is better than C2.qi; ‘Being better’ may mean ‘<‘ (e.g., waste) or ‘>’ (e.g., profit); If C1 dominates C2, this no longer needs to be true if we add a further cat.-II quantity; The more cat.-II quantities, the fewer dominated solutions. q2 (e.g., waste) C2 C1 dominates C2 C1 C1 dominates C3 C2,C3: no dominance q1(e.g., profit) ‘Dominance’ means: being better in all respects. For design, this means: the artefact being better w.r.t. all (properties of all) stakeholders’ values.

  29. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    29 Cat.-II –space and dominance Only non-dominated solutions are relevant  dominance allows pruning cat.-I space; Since nr. non-dominated solutions is smaller with more cat.-II quantities, nr. of cat.-II quantities should be small; For 2 cat.-II quantities, the cat.-II space can be visualized; Dominance is defined, however, for any nr. cat.-II quantities. Dominance is a simple criterion to prune cat.-I space. We only need to consider non-dominated solutions. The relative reduction is larger with fewer cat.-II quantities.

  30. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    30 Trade-offs and the Pareto front In cat.-II space, dominated areas are half-infinite regions bounded by iso-coordinate lines/planes; Solutions falling in one of these regions are dominated and can be ignored in cat.-I-space exploration; Non-dominated solutions form the Pareto front. D Cat.-II quantities f1 and f2 both need to be minimal. A and B are non-dominated, C is dominated. Of A and B, none is better in absolute sense. Solution D would dominate all other solutions – if it would exist.

  31. direction of absolute deterioration tangent to the pareto-front: trade-offs direction of absolute improvement Bonus: when we apply a monotonous mapping to some or all cat.-II quantities, the collection non-dominated solutions stays the same. Example: it doesn’t matter if a penalty function is |a-b| or (a-b)2 A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Output of the Functional Model: Category II    31 Trade-offs and the Pareto front Relevance of Pareto-front: • it bounds the achievable part of cat.-II space; • solutions not on the Pareto front can be discarded; • it exists for any model function, although in general it can only be approximated by a disjoint collection of solutions; • if (part of) it is smooth, it defines two directions in cat.-II space: the direction of absolute improvement / deterioration, and the plane perpendicular to this direction which is tangent to the Pareto front and represents local trade-off relationships between cat.-II quantities. Cat.-II quantities f1 and f2 both need to be minimal. A and B are non-dominated, C is dominated. Of A and B, none is better in absolute sense.

  32. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Limitations from Context: Category III    32 In order to evaluate a model function, we may need quantities, not in category I; These are category-III quantities from the model context, not modifiable by the modeler; Example: legislature, demography, physics, economy, vendor catalogues, human conditions, … Challenge the demarcation between cat.-I and cat.–III for innovative design. Example: when designing thermal house insulation, heat leakage through the windows occurs in the design function. If the window area is in cat.-I, zero-sized windows might be optimal. Else the window size is in cat.-III.

  33. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Intermediate Quantities: Category IV    33 Start the construction of the model by introducing cat.-II; Quantities that don’t depend on anything are cat.-I or cat.-III quantities; All other quantities are cat.-IV quantities. A visual impression of the design function. Green: cat.-I; grey: cat.-II; yellow: cat.-III; blue: cat.-IV. Points represent quantities, not values Arrows indicate functional dependency; notice: no cycles! The entire network is constructed using the scheme of week 4, starting with cat.-II. When the to-do list is empty, all quantities are defined in terms of cat.-I and cat.-III quantities only.

  34. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Optimality and Evolution    34 Our mission is to find ‘good’ or even ‘best’ concepts in cat.-I space. Mathematical optimization regards single-valued functions; Approach typically imitates a mountaineer climbing to the top of a (single-valued) mountain; This would correspond to the situation of a single cat.-II quantity, or all cat.-II quantities lumped; We seek something more generic. Mathematical optimization attempts to find a local or even global extreme of a single-valued function. Most methods work by iteration, i.e., following a mountaineer on its route to the top. This approach would only apply to model functions in case of 1 cat.-II quantity.

  35. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    35 Idea (Eckart Zitzler): combine Pareto and Evolution. Main features of evolution: • genotype encodes blueprint of individual (‘cat.-I’); • genotype is passed over to offspring; • new individual: genotype  phenotype, determining its fitness (‘cat.-II’); • variations in genotypes (mutation, cross-over) cause variation among phenotypes; • fitter phenotypes have larger change of surviving, procreating, and passing their genotypes on to next generation. Evolution as a principle for development may occur in biological and artificial systems alike

  36. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    36 Idea (Eckart Zitzler): combine Pareto and Evolution. ‘Strength’ (hence ‘SPEA’) is a property of an ATBD, derived from the cat.-II quantities, indicating how few it is dominated by, i.e. how fit it is. Issues to resolve: • How to start  population of random individuals (tuples of values for cat.-I quantities); • How to define fitness  fitter when dominated by fewer; • Next generation  preserve non-dominated ones; complete population with mutations and crossing-over; • Convergence  if Pareto front no longer moves. Charles Darwin Pareto and Darwin: the dynamic duo of optimal design (under direction of E. Zitzler)

  37. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    37 Idea (Eckart Zitzler): combine Pareto and Evolution. Caveats: Pareto-Genetic is not perfect • If the fraction dominated concepts is too large, evolution ceases to make progress; • If there broad niches, finding individuals in a narrow niche may be problematic; • Approximations may fail to get anywhere near the theoretical best Pareto front. (No guarantee that analytical alternatives exist) DON’T use Pareto-Genetic if guarantee for optimal solution is required. Charles Darwin Nothing is perfect. There are cases when Pareto-genetic optimization does not meet its target, or when it should not be used.

  38. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    38 demo Charles Darwin

  39. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Approximating the Pareto Front    39 Idea (Eckart Zitzler): combine Pareto and Evolution. If anything else fails: • Complementary approach: local optimization (to be applied on all elements of the Pareto-front separately); • Split cat.-I space in sub spaces if model function behaves different in different regimes (e.g., too much cat.-I freedom may lead to bad evolution progress); • Temporarily fix some cat.-IV quantities (pretend that they are in category-III). Charles Darwin If anything else fails, there are few brute-force methods that may help in difficult situations

  40. A Core Course on Modeling Week 5-Roles of Quantities in a Functional Model    Summary    40 • 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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