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Understanding Cat-II Space and Dominance: Trade-offs and the Pareto Front in Modeling

This course provides a comprehensive introduction to modeling, focusing on Cat-II spaces and dominance. Participants will learn to formulate a purpose, identify entities, choose relations, and formalize models to obtain and interpret results. Emphasizing optimization with multiple criteria, the course will explore challenges such as lumping penalties and the relevance of non-dominated solutions. Key concepts include trade-offs and the Pareto front, highlighting the critical importance of identifying effective solutions while pruning dominated areas within Cat-II space.

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Understanding Cat-II Space and Dominance: Trade-offs and the Pareto Front in Modeling

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  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.24

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

  3. 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

  4. 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

  5. Cat.-II –spaceanddominance A.f1<C.f1 and A.f2<C.f2, so Adominates C A.f2<B.f2 and B.f1<A.f1, so B and A don’tdominateeachother f1 shouldbeminimal (e.g., costs) B.f1<C.f1 and B.f2<C.f2, so B dominates C f2 shouldbeminimal (e.g., waste)

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

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

  8. 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

  9. 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?

  10. 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

  11. 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.

  12. 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.

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

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

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

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

  17. 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,

  18. 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,

  19. 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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