1 A Core Course on Modeling      Contents      What is a Formal Model? A Practical Route to Formal Models Example 1: The Detergent Problem Example 2: The Chimney Sweepers Problem Example 3: The Peanut Butter Problem The relation wizard The function selector Summary References to lecture notes + book References to quiz-questions and homework assignments (lecture notes) Week 4-The Function of Functions

2 A Core Course on Modeling      What is a Formal Model?      What is the meaning of + ? resistors: R tot = R 1 + R 2 springs: C tot = 1/(1/C 1 +1/C 2 ) resistors: R tot = 1/(1/R 1 + 1/R 2 ) springs: C tot = C 1 + C 2 the intuition of ‘addition’, ‘accumulation’: Week 4-The Function of Functions

3 A Core Course on Modeling      What is a Formal Model?      What is the meaning of + ? Week 4-The Function of Functions Conclusion: there is always a need for interpretation going from relations in ‘real world’ to mathematical relations There is no simple, generic way to infer mathematical relations from relations in real world We need a (heuristic) process to do so. the intuition of ‘addition’, ‘accumulation’:

4 A Core Course on Modeling      A Practical Route to Formal Models      Heuristics to arrive at formal expressions: meaningful names chain of dependencies todo-list dimensional analysis wisdom of the crowds two models is better than one model when is a model good enough? Baron von Münchhausen In a conceptual model, properties are always part of a concept (‘myCar.wheel.diameter’). In a formal model, properties may be ‘just’ quantities (‘myCarWheelDiameter) with names that may be meaningfully abbreviated (‘mCWhD’). General scheme: start with the quantitiy needed for your purpose, and try to express this in other quantities in the simplest possible form with as few as possible assumptions such that the quantity is written as function of the other quantities continue with the arguments of the function Everytime when introducing a new quantity, add it to the todo list. When a quantity is expressed into in something known (= a value or another expression), take it off the todo list. When the todo list is empty, model’s first version is ready. When seeking a mathematical expression, if possible, use dimensional considerations to find needed expression … … but at least verify expression with dimensional analysis … even if this may require inventing units and dimensions. Often values need guessing. If you can involve a number of people, let them guess independently. This gives (a) a more accurate estimate if there are no systematic errors and (b) an idea of the variation. Otherwise: try to relate the unknown values to values you (and your friends) may know. Sometimes, there are two or more routes to (part) of your model. Implement them all, and compare the results. The spread in results is a clue to the reliability of the achievable outcome. A model is never complete and fully accurate. But, given its purpose, it can be complete and accurate enough. (See chapter 6.) Regularly check if your purpose is met. Once you have your first version running, you can: use it to find out which of the uncertainties of your inputs are the most dominant  try to get these more accurate if necessary find out which modifications could be worthwhile Week 4-The Function of Functions

5 A Core Course on Modeling      The Detergent Problem      “What is the total amount of detergent annually dumped in the Environment in the Netherlnds? Week 4-The Function of Functions

6 A Core Course on Modeling amAnDetDmp = f(nrAnWshs, detPWsh) relationsdimensionsassumptionstodo [kg / year] = F([wash / year], [kg / wash]) amAnDetDmp nrAnWshs detPWsh Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump      The Detergent Problem      Week 4-The Function of Functions

7 A Core Course on Modeling amAnDetDmp = nrAnWshs * detPWsh relationsdimensionsassumptionstodo [kg / year] = [wash / year] * [kg / wash] amAnDetDmp nrAnWshs detPWsh Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump      The Detergent Problem      Week 4-The Function of Functions

8 A Core Course on Modeling amAnDetDmp = nrAnWshs * detPWsh relationsdimensionsassumptionstodo [kg / year] = [wash / year] * [kg / wash ] Washing laundry is the only way detergent gets into the environment amAnDetDmp nrAnWshs detPWsh nrAnWshsPFam nrFamIH nrPIH nrPPFam nrAnWshs = nrAnWshsPFam * nrFamIH [wash / year] = [wash / (fam *year)] * [fam] No institutional laundry washing, only families nrFamIH = nrPIH / nrPPFam [fam] = [people] / [people / fam] Everybody belongs to exactly one family: families are disjoint nrPIH = 17  0.5 million [people] common knowledge nrPPFam = 2.2  0.2 [people/fam] public domain nrAnWshsPFam = 100  20 [wash / year] wisdom of the crowds Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump detPWsh = 0.17  0.03 [kg / wash] wisdom of the crowds todo list is empty  model is ready      The Detergent Problem      Week 4-The Function of Functions

9 A Core Course on Modeling todo amAnDetDmp nrAnWshs detPWsh nrAnWshsPFam nrFamIH nrPIH nrPPFam      The Detergent Problem      Week 4-The Function of Functions amAnDetDmp nrPPFam nrPIH nrFamIH nrAnWshsPFam detPWsh nrAnWshs * / *

10 A Core Course on Modeling todo amAnDetDmp nrAnWshs detPWsh nrAnWshsPFam nrFamIH nrPIH nrPPFam      The Detergent Problem      Week 4-The Function of Functions amAnDetDmp nrPPFam nrPIH nrFamIH nrAnWshsPFam detPWsh nrAnWshs * / * 17  0.5 million 2.2   … million 0.17   … million 772  …million 100  20

11 A Core Course on Modeling This type of model is a ‘thumbnail calculation’: OK for quick order of magnitude estimations Uses straightforward substitutions, only based on dimension analysis Work with intervals to get an impression of the variation of the answers (143  67 M; correct value according to various sites such as is 150 M kg) Purpose: pub quizzes, trivial pursuit, … a.k.a. ‘sledgehammer estimation’      The Detergent Problem      Week 4-The Function of Functions

12 A Core Course on Modeling About accuracy model: outcome = function of inputs y=f(x 1,x 2,…)  y=  f/  x 1  x 1 +  f/  x 2  x 2 + … So: |  y|=|  f/  x 1 | |  x 1 | + |  f/  x 2 | |  x 2 | + … This is a very pessimistic upperbound: all quantities need to conspire to give worst case deviation. In chapter 6, we will get a more realistic estimate.      The Detergent Problem      Week 4-The Function of Functions

13 A Core Course on Modeling      The Chimney Sweepers Problem      “How many chimney sweepers work in Amsterdam?” Week 4-The Function of Functions

14 A Core Course on Modeling nrChSwIA = nrChIA * nrSwPCh relationsdimensionsassumptionstodo [Sw / A] = [Ch / A] * [SW / Ch] Amsterdam ch.-sweepers sweep Amsterdam chimneys only nrChSwIA nrChIA nrSwPCh nrChPFam nrFamIA nrPIA nrPPFam nrChIA = nrChPFam * nrFamIA [Ch / A] = [Ch / Fam] * [Fam / A] ch.-sweepers sweep only chimneys on family houses nrFamIA = nrPIA / nrPPFam [Fam / A] = [P / A] / [P / Fam] nrPPFam is the same everywhere (does not depend on ‘Amsterdam’) nrPIA = [P] common knowledge nrPPFam = 2.2  0.2 [P/Fam] public domain nrChPFam (= 1/nrFamPCh) =0.1  0.02 [Ch/Fam] wisdom of the crowds Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;      The Chimney Sweepers Problem      Week 4-The Function of Functions

15 A Core Course on Modeling nrSwPCh = nrSwPSe * nrSePCh relationsdimensionsassumptionstodo [Sw / Ch] = [Sw*year/Se] * [Se/(Ch*year)] Introduce time to associate sweeper’s ca- pacity to chimney’s need nrChSwIA nrChIA nrSwPCh nrChPFam nrFamIA nrPIA nrPPFam nrSwPSe nrSePCh timeP1Se timeP1Sw nrSwPSe = timeP1Se / timeP1Sw [Sw * year / Se] = [hour / Se] / [hour / (Sw*year)] assume average times (i.e., no season influences etc.) timeP1Sw = 1200  100 hour / Sw * year) Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service; timeP1Se = 2  0.25 hour / Se wisdom of the crowds work year = 1600 hours nrSePCh = 1 Se /( Ch * year) insurance requirement todo list is empty  model is ready but what does it mean ?  NOTHING, since we formulated no purpose      The Chimney Sweepers Problem      Notice: two different units, both with dimension time. To verify that units are consistent, substitute back into expressions for nrSwPSe and nrSwPCh: check that all units multiply and divide to produce the correct final result. In this case: nrSwPCh=timeP1Se * nrSePCh / timeP1Sw has unit ‘sweeper / chimney’ – which is correct. Week 4-The Function of Functions To find an expression for nrSwPCh, ask: ‘what links the nr of sweepers to the nr chimneys?’. Answer: sweepers service chimneys. ‘How many services’ (1) relates to the capacity (=available resource) of a sweeper, and (2) to the need of a chimney (=needed resource). Here, ‘resource’=time. The capacity of a sweeper is expressed in the time he works ([Sw*year]); The ‘need’ of a chimney is therefore expressed ([Ch*year]).

16 A Core Course on Modeling but what does it mean ?  NOTHING, since we formulated no purpose What purposes could we think of: are there at least 300 Chimney Sweepers so that we can begin a professional journal? so: we only need to know if NrChSwIA > 300 are there less than 50 Chimney Sweepers so that we can have next year’s ChSw convention meeting in the Restaurant ‘the Swinging Sweeper?’ so: we only need to know if NrChSwIA <50 are there about as many Chimney Sweepers as there are Sewer Cleaners so that we can form efficient ‘Chimney and Sewage Control and Service Units’? so: we only need to know if NrChSwIA is between 50 and 60… … each purpose poses different challenges / allows different approximations in our model.      The Chimney Sweepers Problem      Week 4-The Function of Functions

17 A Core Course on Modeling To assess credibility of a model: confront with actual measurements confront with outcome of a second, independent model how much soot and ashes are disposed of by the municipal Ash & Soot Depot? how often do you see a chimney sweeper at work (wisdom of the crowds)? how much money do people in A. spend annually in cleaning their chimneys? check out a model for the later case: (notice: this model is not completely independent from the previous; it uses some common quantities)      The Chimney Sweepers Problem      doing experiments: beyond the scope of modeling Week 4-The Function of Functions

18 A Core Course on Modeling Thusfar, all mathematical relations were obtained via dimension analysis. Dimensions are more generic than just SI dimensions/units. Dimensional analysis often works for models of a particular form: y = x 1 n1 * x 2 n2 * x 3 n3 … =  i x i ni, where (integer) n i can be both larger and smaller than 0. How to go about when other forms are needed?      The Chimney Sweepers Problem      Week 4-The Function of Functions

19 A Core Course on Modeling      The Peanut Butter Problem      “How to get rich by selling a new brand of peanut butter?” PB Week 4-The Function of Functions

20 A Core Course on Modeling income = pricePerItem * nrSoldItems relationsdimensionsassumptionstodo [Euro / year] = [Euro/myPB] * [myPB/year] no discount with larger quantities per purchase profit income expenses pricePerItem nrSoldItems nrSoldTotal marketShare nrSoldItems = nrSoldTotal * marketShare [myPB / year] = [allPB / year] * [myPB/allPB] my PB will not increase the total market pricePerItem = … nrSoldTotal = … [allPB/year] from a neutral marketing bureau marketShare = … problems: I have a choice for pricePerItem marketShare depends on pricePerItem      The Peanut Butter Problem      Week 4-The Function of Functions profit = income - expenses[Euro / year] = [Euro/year] no taxes, no inflation

21 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 1: glass box (glass jar … ;-) : pricePerItem marketShare 0 1 ?      The Peanut Butter Problem      Week 4-The Function of Functions

22 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 1: glass box (glass jar … ;-) : pricePerItem marketShare 0 1 What mechanism determines marketShare(pricePerItem)? monotonically decreasing between 0 and 1 asymptote: marketShare  0 if pricePerItem   asymptote: marketShare  1 if pricePerItem  -  what sort of mathematical dependency ???      The Peanut Butter Problem      If something gets more expensive, the chance people will buy it decreases A market share cannot be 1 but only if it creates additional request If something gets sufficiently expensive, nobody will buy it If something gets sufficiently cheap, every potentially interested customer will buy (or get!) it It would need a glass box model of customers’ brains to derive this dependency from first principles Week 4-The Function of Functions

23 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 1: glass box (glass jar … ;-) : pricePerItem marketShare 0 1 cheapest competitor most expensive competitor 1st guess: straight lines advantage: simple disadvantage: not smooth uncertain: does this represent the actual behavior?      The Peanut Butter Problem      Week 4-The Function of Functions

24 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 1: glass box (glass jar … ;-) : pricePerItem marketShare 0 1 cheapest competitor most expensive competitor 2nd, 3rd guess: arctan, logistic, …? advantage: smooth disadvantage: more parameters? what values for the parameters? uncertain if this follows the actual behaviour      The Peanut Butter Problem      Depending on what you are going to DO with the math (e.g., optimisation), smoothness can be important Week 4-The Function of Functions

25 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 1: glass box (glass jar … ;-) : pricePerItem marketShare 0 1 cheapest competitor most expensive competitor      The Peanut Butter Problem      Week 4-The Function of Functions

26 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 2: black box: use panel of test subjects; ask them if they would buy your PB for price X pricePerItem marketShare 0 1 do a smooth curve fit (e.g., splines) that satisfies 0  marketShare  1      The Peanut Butter Problem      Week 4-The Function of Functions

27 A Core Course on Modeling problems: I have a choice for pricePerItem marketShare depends on pricePerItem Approach 2: black box: use panel of test subjects; ask them if they would buy your PB for price X pricePerItem marketShare 0 1 do a smooth curve fit (e.g., splines) that satisfies 0  marketShare  1 but, anyway: let us assume we have some function: then the model predicts the income as function of the pricePerItem.      The Peanut Butter Problem      Week 4-The Function of Functions

28 A Core Course on Modeling Revisit the peanut butter example: income=pricePerItem * nrSoldItems nrSoldItems=nrSoldTotal * marketShare marketShare=f(pricePerItem) for convenience, introduce abbreviations:      The Peanut Butter Problem      Week 4-The Function of Functions

29 A Core Course on Modeling Revisit the peanut butter example: inc =pPI * nSI (inc=income; pPI=pricePerItem; nSI=nrSoldItems) nSI = nST * mSh (nST=nrSoldTotal; mSh=marketShare) mSh=f(pPI) So: inc = pPI * nST * mSh(pPI) This is naive: first, realize that nST =  i nST i, i ranges over customers. So: inc = pPI * mSh(pPI) *  i nST(i) Improved model: inc = pPI *  i nST(i) * mSh(pPI, i)      The Peanut Butter Problem      This is irrelevant if all customers would buy equal amounts of peanutbutter. But: there are mega- consumers and mini-consumers ! Realize that the decision to choose MY peanutbutter is taken by a customer. Suppose that the majority of mega customers would decide against my peanutbutter – then the original model with mSh(pPI) instead of mSh(pPI,i) would give an overestimate  misleading. So: think about which arguments a function should depend on! Week 4-The Function of Functions

30 A Core Course on Modeling To go from conceptual model to formal model: while your purpose is not satisfied: 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 ready      The Peanut Butter Problem      Week 4-The Function of Functions

31 A Core Course on Modeling To go from conceptual model to formal model: while your purpose is not satisfied: 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 ready spline arctan asymptote function selector relation wizard logistic function o p t i m i s a t i o n      The Peanut Butter Problem      Week 4-The Function of Functions

32 A Core Course on Modeling Conceptual model  formal model : not in a formally provable correct way; Appropriate naming Structure Chain of dependencies: the formal model as a directed acyclic graph; What mechanism? What quantities drive this mechanism? What is the qualitative behavior of the mechanism? What is the mathematical expression to describe this mechanism? To-do-list : all intermediate quantities are found and elaborated in turn; Formation of mathematical expressions: dimensional analysis  mathematical expressions, e.g in the case of proportionality the Relation Wizard can help finding appropriate fragments of mathematics; the Function Selector can help finding an appropriate expression for a desired behavior; wisdom of the crowds can help improve the accuracy of guessed values;      Summary      Week 4-The Function of Functions