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The art of Devising Thumnail Models

The art of Devising Thumnail Models. Kees van Overveld for Marie Curie International PhD exchange Program. http://www.sxc.hu/photo/1361109. “How many chimney sweepers work in Eindhoven?”. relations. dimensions. assumptions. todo. Eindhoven ch.-sweepers sweep Eindhoven chimneys only.

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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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