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1 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 Summary References to lecture notes + book References to quiz-questions and homework assignments (lecture notes) Week 5-Roles of Quantities in a Functional Model

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

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

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4 A Core Course on Modeling the printers 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, …) …. but do we have 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

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5 A Core Course on Modeling Unclarity about what depends on what is a main source of confusion in functional models. the printers dilemma: reading light, reading easy or reading much? 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

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6 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 (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

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7 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 pick C from to do list: C depends on T, S 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 blue, underlined quantities appear underway to express what quantities depend on

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8 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 = 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

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9 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 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

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10 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

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11 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

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12 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 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

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13 A Core Course on Modeling categorydepends onmeaningtypeexample I quantity that can be freely modified nothing modelers decisions, modifications, interventions, explorations … any physical dimensions, options, tweakable parameters, unknowns II quantity that expresses the need (purpose) of the model I,III,IV modelers goals (purpose) often: ordinal (decide, optimize, steer/control, …) profit, comfort, safety, …things for interest of the stakeholder III quantity from context (not freely modifiable) nothing beyond the authonomy of the modeler any physical constants, vendors catalogue data, … IVauxiliary, intermediate quantity I,III,IVinternal – only needed to execute the model; values are ultimately irrelevant any Week 5-Roles of Quantities in a Functional Model The 4-Categories Approach

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

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15 A Core Course on Modeling Input of the Functional Model: Category I 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, … Week 5-Roles of Quantities in a Functional Model

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16 A Core Course on Modeling Category-I quantities correspond to independent, free decisions / modifications / explorations / …. The printers dilemma: T, S and P can not all be in category I, since TS 2 /P=constant. Choosing appropriate cat.-I quantities may require cutting the Gordian knot. 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.

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

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18 A Core Course on Modeling 1.Dont include too many cat.-II quantities; 2.Include the right cat.-II quantities; 3.Cat.-II quantities for design etc. must be ordinal; 4.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. Week 5-Roles of Quantities in a Functional Model Output of the Functional Model: Category II

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

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20 A Core Course on Modeling Dilemma: many/few cat.-II quantities? consider the book printers example: three models cat.-I: S,T; cat.-II: P=TS 2 /A; q P = max(P-P 0,0) cat.-I: T,P; cat.-II: S= PA/T; q S = - min(S-S 0,0) cat.-I: P,S; cat.-II: T= PA/S 2 ; q T = - min(T-T 0,0) In each model, q i expresses something that is unwanted: the smaller q i, the better. The q i punish unwanted behavior: penalty functions. If nr. pages is larger than P 0, q P is larger than 0. If point size is less than S 0, q S is larger than 0 Week 5-Roles of Quantities in a Functional Model reading light: reading easy: reading much: If text is less than T 0, q T is larger than 0 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

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21 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 (use function selector to find suitable penalty!) Week 5-Roles of Quantities in a Functional Model Output of the Functional Model: Category II

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22 A Core Course on Modeling Dilemma: many/few cat.-II quantities? Penalty function: the smaller the better. Every q i is a cat-II quantity, associated to a desired condition. Adding penalty functions: Q= i q i, to express that multiple conditions should hold simultaneously. For Q: the smaller the better. If separate q i 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. Week 5-Roles of Quantities in a Functional Model Output of the Functional Model: Category II

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23 A Core Course on Modeling Dilemma: many/few cat.-II quantities? However: adding penalty functions may violate dimension constraints; adding penalty functions introduces (arbitrary) weights: Q= i a i q i, even if the a i 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! Week 5-Roles of Quantities in a Functional Model Output of the Functional Model: Category II

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

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

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

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27 A Core Course on Modeling Cat.-II –space and dominance Assume cat.-II quantities are ordinals: Every axis in cat.-II space is ordered; concept C 1 dominates C 2 iff, for all cat.-II quantities q i, C 1.q i is better than C 2.q i ; Being better may mean (e.g., profit); If C 1 dominates C 2, 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. Week 5-Roles of Quantities in a Functional Model Output of the Functional Model: Category II

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28 A Core Course on Modeling Cat.-II –space and dominance Assume cat.-II quantities are ordinals: Every axis in cat.-II space is ordered; concept C 1 dominates C 2 iff, for all cat.-II quantities q i, C 1.q i is better than C 2.q i ; Being better may mean (e.g., profit); If C 1 dominates C 2, 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. 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

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

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30 A Core Course on Modeling 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. 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. D Solution D would dominate all other solutions – if it would exist. Week 5-Roles of Quantities in a Functional Model Output of the Functional Model: Category II

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31 A Core Course on Modeling 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. 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 Bonus: when we apply a monotonous mapping to some or all cat.-II quantities, the collection non-dominated solutions stays the same. Example: it doesnt matter if a penalty function is |a-b| or (a-b) 2

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32 A Core Course on Modeling Limitations from Context: Category III 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. Week 5-Roles of Quantities in a Functional Model

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33 A Core Course on Modeling Intermediate Quantities: Category IV Start the construction of the model by introducing cat.-II; Quantities that dont 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. Week 5-Roles of Quantities in a Functional Model

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34 A Core Course on Modeling Optimality and Evolution 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. Week 5-Roles of Quantities in a Functional Model

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35 A Core Course on Modeling Approximating the Pareto Front 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 Week 5-Roles of Quantities in a Functional Model

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36 A Core Course on Modeling Idea (Eckart Zitzler): combine Pareto and Evolution. 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) 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. Approximating the Pareto Front Week 5-Roles of Quantities in a Functional Model

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37 A Core Course on Modeling Idea (Eckart Zitzler): combine Pareto and Evolution. Caveats: Pareto-Genetic is not perfect If the fraction non-dominated concepts is too large, evolution makes no 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) DONT 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. Approximating the Pareto Front Week 5-Roles of Quantities in a Functional Model

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

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39 A Core Course on Modeling 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 Approximating the Pareto Front Week 5-Roles of Quantities in a Functional Model resized-600.jpg

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40 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

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