A core Course on Modeling

1. A core Course on Modeling Introductionto Modeling 0LAB0 0LBB0 0LCB0 0LDB0 c.w.a.m.v.overveld@tue.nl v.a.j.borghuis@tue.nl S.22

2. define formulate purpose identify entities choose relations conceptualize formalize relations obtain values formalize operate model obtain result execute present result interpret result conclude Executionphase: operate model

3. Category-I quantitiescorrespondto free decisions/ modifications / explorations / …. Category-II quantitiescorrespondtothingsyou want. But whatdoyou want? http://www.morguefile.com/archive/display/156086 s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

4. Category-I quantitiescorrespondto free decisions/ modifications / explorations / …. Category-II quantitiescorrespondtothingsyou want. But whatcouldyou want? http://www.morguefile.com/archive/display/156086 s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

5. Category-I quantitiescorrespondto free decisions/ modifications / explorations / …. Category-II quantitiescorrespondtothingsyou want • nothing: everychoice is good • W minimal: make s=0 • s minimal: make s=0 • t minimal: make s=0 not veryinteresting s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

6. Category-I quantitiescorrespondto free decisions/ modifications / explorations / …. Category-II quantitiescorrespondtothingsyou want not veryinteresting • W minimaland t minimal: s=0 • s maximaland t minimal: v • W minimaland s maximal: interesting • W minimaland s maximaland t minimal: interesting • interesting cases involve>1 criterion • ...but not the other way round s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

7. Category-I quantitiescorrespondto free decisions/ modifications / explorations / …. Category-II quantitiescorrespondtothingsyou want • 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 s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

8. Category-I quantitiescorrespondto free decisions/ modifications / explorations / …. Category-II quantitiescorrespondtothingsyou want 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 s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

9. 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 effort: qW=|W(t,v)| = W(t,v) large distance: qs=|s-1(t,v)| s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

10. 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 unit of ws is km unit of ww is Joule-1 examples: little effort and large distance: Q = wwqW+wsqs =ww W(t,v)+ws|s-1(t,v)| little effort and large distanceandlittle time: Q = wwW(t,v)+ws|s-1(t,v)|+wtt s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density unit of wt is hour-1

11. 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) s = distancetravelled t = duration of the tour v = velocity W = performedwork Fw = wind force c = constant A = area  = air density

12. 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

13. 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

14. 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

15. 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

16. Summary: formulateproblem in terms of criteria criteria correspondto cat.-II quantities considerexpressing criteria as penalties: quantities q, q0,thatshouldbe small as possible choose criteria suchthat non-trivialproblemresults multiple criteria: consideraddingpenalties Q=iwiqiwith proper weightswi, wi>0 form expressions with |...|, max(...) etc. toexpress right type of criterion