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The art of Devising Thumnail Models

This study delves into the complexities of modeling the number of chimney sweepers in Eindhoven within the context of the Marie Curie International PhD exchange program. We explore various assumptions, relations, and dimensions associated with chimney sweeping services, while also emphasizing the importance of purpose-driven inquiries. By employing a structured approach, we analyze the interplay between family households and chimney sweepers to derive meaningful insights and facilitate further research or professional engagement in the community.

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The art of Devising Thumnail Models

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  1. The art of Devising Thumnail Models Kees van Overveld for Marie Curie International PhD exchange Program

  2. http://www.sxc.hu/photo/1361109 • “How manychimneysweepers • work in Eindhoven?”

  3. relations dimensions assumptions todo Eindhoven ch.-sweepers sweep Eindhoven chimneys only nrChSwIE = nrChIE * nrSwPCh [Sw / E] = [Ch / E] * [SW / Ch] nrChSwIE nrChIE nrSwPCh nrChPFam nrFamIE nrPIE nrPPFam ch.-sweepers sweep only chimneys on family houses nrChIE = nrChPFam * nrFamIE [Ch / E] = [Ch / Fam] * [Fam / E] nrPPFam is the same everywhere (does not depend on ‘Eindhoven’) nrFamIE = nrPIE / nrPPFam [Fam / E] = [P / E] / [P / Fam] common knowledge nrPIE = 250000 [P] public domain nrPPFam = 2.2  0.2 [P/Fam] nrChPFam (= 1/nrFamPCh) =0.10.02 [Ch/Fam] wisdom of the crowds • Sw=sweeper; Ch=chimney;E=Eindhoven;Fam=Family;P=people;Se=Service;

  4. relations dimensions assumptions todo relate sweeper’s capacity to chimney’s need nrSwPSe * nrSePCh nrSwPCh = [Sw / Ch] = [Sw*year/Se] * [Se/(Ch*year)] nrChSwIE nrChIE nrSwPCh nrChPFam nrFamIE nrPIE nrPPFam nrSwPSe nrSePCh timeP1Se timeP1Sw assume average times (i.e., no season influences etc.) nrSwPSe = timeP1Se / timeP1Sw [Sw * year / Se] = [hour / Se] / [hour / (Sw*year)] wisdom of the crowds timeP1Se = 20.25 hour / Se work year = 1600 hours timeP1Sw = 1200100 hour / Sw * year) insurance requirement nrSePCh = 1 Se /( Ch * year) todo list is empty  model is ready • Sw=sweeper; Ch=chimney;E=Eindhoven;Fam=Family;P=people;Se=Service;

  5. QUIZ Whydid we compute the number of chimneysweepers in Eindhoven? http://www.sxc.hu/browse.phtml?f=download&id=587687

  6. Where does this lead to? Nowhere ...since we formulated no purpose! http://www.sxc.hu/browse.phtml?f=download&id=587687

  7. Whatpurposescould we thinkof? QUIZ Give at leastthree different reasonswhy we couldbeinterested in the number of chimneysweepers in Eindhoven?

  8. Whatpurposescould we think of: e.g. verification • are there at least 300 ChimneySweeperssothat we can begin a professional journal? so: we onlyneedtoknowifNrChSwIE > 300

  9. Whatpurposescould we think of: e.g. verification • are therelessthan 50 ChimneySweeperssothat we can have next year’sChSwconvention meeting in the Restaurant ‘the Swinging Sweeper?’ so: we onlyneedtoknowifNrChSwIE <50

  10. Whatpurposescould we think of: e.g. verification • are thereabout as manyChimneySweepers as there are SewerCleanerssothat we can form efficient ‘ChimneyandSewage Control and Service Units’? so: we onlyneedtoknowifNrChSwIE is between 20 and 30

  11. Whatpurposescould we think of: • … eachpurpose • poses different challenges • allows different approximations

  12. How did the construction of the model go about? quantities we are free todecide quantitiesto help usexpressingdependencies a quantity we wereinterested in Cat I:choice quantities we guess (measure, look up, ...) from the problem context Cat IV: intermediate Cat II: objectives Cat III: context

  13. Tryityourself: make a model for a goodmeal • whatconstitutes 'good'? • how is thisexpressed in terms of cat-II quantities? • what relations governtheirvalues? • whatifsomequantities are not numbers? a model for good meal a model for good and a cheap meal

  14. Category-I quantitiescorrespondto • free decisions/ modifications / explorations / …. • Category-II quantitiescorrespondtothingsyou want as a result of these decisions • minimalprice don'teat • minimal cholesterol andminimalprice:  don'teat • enoughkCal  eatenoughbread • minimal cholesterol andenoughkCal: interesting • enoughkCaland as cheap as possible: interesting • interesting cases involve>1 criterion • ...but not the other way round not veryinteresting

  15. Category-I quantitiescorrespondto • free decisions/ modifications / explorations / …. • Category-II quantitiescorrespondtothingsyou want as a result of these decisions • conclusion: • manyoptimizationproblemsinvolvetrade-offs • examples: • largest volume with smallest area • largestprofit with smallest investment • largest ... with least ... • largestvelocity with largestsafety • largest ... with largest ... (andothercombinations) • also cases with >2 criteria oftenoccur. http://www.morguefile.com/archive/display/93433

  16. Category-I quantitiescorrespondto • free decisions/ modifications / explorations / …. • Category-II quantitiescorrespondtothingsyou want as a result of these decisions QUIZ In order towrapsomethingefficiently, I seekfor a shape with maximal volume andminimal area. In what case could I want a shape with minimal volume andmaximal area? • conclusion: • manyoptimizationproblemsinvolvetrade-offs • examples: • largest volume with smallest area • largestprofit with smallest investment • largest ... with least ... • largestvelocity with largestsafety • largest ... with largest ... (andothercombinations) • also cases with >3 criteria oftenoccur. http://www.morguefile.com/archive/display/93433

  17. Toexpress criteria, usepenalties. • A penalty q = f(cat.-I quantities) is • a cat.-II quantity • a function of cat.-I quantities • 0 (q=0 is ideal) • shouldbe as small as possible http://commons.wikimedia.org/wiki/File:Europe_punishes_the_spoilt_kid_(Greece)_for_asking_too_much.jpg • examples: • little cholesterol: qC=|amountChol| = amountChol • enoughkCal: qK=|amountKCal - optimalAmountKCal|

  18. Toexpress criteria, usepenalties. • multiple criteria: multiple penalties • add: Q= iqi onlyifqi have samedimension • Q= iwiqi ,wi>0 wiqi must have samedimension • weightswi: values ??? • ifwi’> wi, thenqi’ willbe smaller thanqi

  19. Toexpress criteria, usepenalties. • multiple criteria: multiple penalties • add: Q= iqi onlyifqi have samedimension • Q= iwiqi ,wi>0 wiqi must have samedimension • weightswi: values ??? • multiple criteria byaddingpenalties: lumping • advantages: • worksforarbitrarilymany criteria • mayusemathematicaltechniques • for a single Q = f(cat.-I) • (e.g., differentiateandrequirederivativestobe 0)

  20. Toexpress criteria, usepenalties. • multiple criteria: multiple penalties • add: Q= iqi onlyifqi have samedimension • Q= iwiqi ,wi>0 wiqi must have samedimension • weightswi: values ??? • multiple criteria byaddingpenalties: lumping • disadvantages: • whatshouldvaluesforwibe? • addingapplesandorangesmaybe • ethicallyunwanted

  21. Toexpress criteria, usepenalties. • Variationstopenalties (y = f(cat.-I quantities)): • q = y: y shouldbe small; assumethat y0 • q = |y| or q=y2: y shouldbe small in absolute value http://commons.wikimedia.org/wiki/File:Big_and_small_dog.jpg

  22. Toexpress criteria, usepenalties. • Variationstopenalties (y = f(cat.-I quantities)): • q = y: y shouldbe small; assumethat y0 • q = |y| or q=y2: y shouldbe small in absolute value QUIZ What penalty q couldbeusedtoexpressthat y shouldbe smaller thansome y0? http://commons.wikimedia.org/wiki/File:Big_and_small_dog.jpg

  23. Toexpress criteria, usepenalties. • Variationstopenalties (y = f(cat.-I quantities)): • q = y: y shouldbe small; assumethat y0 • q = |y| or q=y2: y shouldbe small in absolute value • q = |max(y,y0)-y0|: y shouldbe smaller than y0 • q = |y0-min(y,y0)|: y shouldbelargerthan y0 • q = |y-y0|: y shouldbe close to y0 • q = 1/|y| or q = 1/(w+|y|), w>0: y shouldbe large • et cetera (usefunctionselector • or imagination!) http://commons.wikimedia.org/wiki/File:Big_and_small_dog.jpg

  24. Cat.-II –spaceanddominance • Revisitoptimization with multiple criteria: • Problems with lumpingpenalties Q=iwiqi • what are the values of wi (trial and error)? • don’taddthingsthatshouldn’tbeadded. •  look foralternative approach, usingdominance

  25. Cat.-II –spaceanddominance • Assume cat.-II quantities are ordinals: • Everyaxis in cat.-II space is ordered; • concept C1dominatesC2 • iff, forall cat.-II quantitiesqi, C1.qi is betterthan C2.qi; • ‘Beingbetter’ maymean ‘<‘ (e.g., waste) or ‘>’ (e.g., profit); http://www.morguefile.com/archive/display/765894

  26. Cat.-II –spaceanddominance A.f1<C.f1 and A.f2<C.f2, so A dominates C A.f2<B.f2 and B.f1<A.f1, so B and A don’t dominate each other f1 should be minimal (e.g., costs) B.f1<C.f1 and B.f2<C.f2, so B dominates C f2 should be minimal (e.g., waste)

  27. Cat.-II –spaceanddominance • Only non-dominatedsolutions are relevant  dominanceallowspruningcat.-I space; http://www.morguefile.com/archive/display/50363

  28. Cat.-II –spaceanddominance • Only non-dominatedsolutions are relevant  dominanceallowspruningcat.-I space; QUIZ http://www.morguefile.com/archive/display/50363 Why does cat.-I spaceneedstobepruned?

  29. Cat.-II –spaceanddominance • Only non-dominatedsolutions are relevant  dominanceallowspruningcat.-I space; • Nr. non-dominatedsolutions is smaller with more cat.-II quantities http://www.morguefile.com/archive/display/50363

  30. Cat.-II –spaceanddominance • Only non-dominatedsolutions are relevant  dominanceallowspruningcat.-I space; • Nr. non-dominatedsolutions is smaller with more cat.-II quantities http://www.morguefile.com/archive/display/50363 QUIZ Why do we have fewer non-dominatedsolutions with more cat.-II quantities?

  31. Cat.-II –spaceanddominance • Only non-dominatedsolutions are relevant  dominanceallowspruningcat.-I space; • Nr. non-dominatedsolutions is smaller with more cat.-II quantitiesnr. of cat.-II quantitiesshouldbesmall (otherwisethere are few dominatedsolutions). http://www.morguefile.com/archive/display/50363

  32. Trade-offsand the Pareto front • In cat.-II space, dominatedareas are half-infiniteregionsboundedbyiso-coordinatelines/planes; • Solutionsfalling in one of these regions are dominatedandcanbeignored in cat.-I-spaceexploration; • Non-dominatedsolutions form the Pareto front.

  33. Trade-offsand the Pareto front D • Cat.-II quantities f1 and f2 bothneedtobeminimal. A and B are non-dominated, C is dominated. Of A and B, none dominates the other.

  34. Trade-offsand the Pareto front D • Cat.-II quantities f1 and f2 bothneedtobeminimal. Solution D woulddominateallothersolutions – ifitwouldexist.

  35. Trade-offsand the Pareto front • Relevance of Pareto-front: • itbounds the achievable part of cat.-II space; • solutions not on the Pareto front canbediscarded.

  36. Trade-offsand the Pareto front • Relevance of Pareto-front: • itexistsforany model function, although in generalitcanonlybeapproximatedby sampling the collectionof solutions.

  37. Trade-offsand the Pareto front • Relevance of Pareto-front: • itdefinestwodirections in cat.-II space:

  38. direction of absolute improvement • Trade-offsand the Pareto front http://cdn.morguefile.com/imageData/public/files/a/alvimann/preview/fldr_2010_03_23/file3831269347533.jpg • Relevance of Pareto-front: • itdefinestwodirections in cat.-II space: the direction of absolute improvement / deterioration,

  39. direction of absolute deterioration • Trade-offsand the Pareto front http://cdn.morguefile.com/imageData/public/files/a/alvimann/preview/fldr_2010_03_23/file3831269347533.jpg • Relevance of Pareto-front: • itdefinestwodirections in cat.-II space: the direction of absolute improvement / deterioration,

  40. tangent to the pareto-front: trade-offs • Trade-offsand the Pareto front http://cdn.morguefile.com/imageData/public/files/a/alvimann/preview/fldr_2010_03_23/file3831269347533.jpg • Relevance of Pareto-front: • itdefinestwodirections in cat.-II space: the direction of absolute improvement / deterioration, and the planeperpendiculartothisdirectionwhich is tangent to the Pareto front, whichrepresentstrade-offsbetween cat.-II quantities.

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