1 / 27

to the Decision Deck platform

2nd Decision Deck Workshop. University Paris Dauphine. February 21-22, 2008. to the Decision Deck platform. UTA GMS /GRIP plugin. Piotr Zielniewicz Poznan University of Technology, Poland. Plan. Problem statement Disaggregation-aggregation (regression) approach UTA GMS method

kamea
Télécharger la présentation

to the Decision Deck platform

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 2nd Decision Deck Workshop University Paris Dauphine February 21-22, 2008 to the Decision Deck platform UTAGMS/GRIP plugin Piotr Zielniewicz Poznan University of Technology, Poland

  2. Plan • Problem statement • Disaggregation-aggregation (regression) approach • UTAGMS method • GRIP method • UTAGMS/GRIP plugin overview • UTAGMS/GRIP plugin demonstration • Conclusions and future works 2nd Decision Deck Workshop, February 21-22, 2008

  3. A x * * x * * * x x x * * x x * * * x x x x x x x * * x x x x Problem statement • Consider a finite set A of alternatives (actions) evaluated by m criteria from a consistent family F ={g1,...,gm} • Taking into account preferences of a Decision Maker (DM), rank all the actions of set A from the best to the worst 2nd Decision Deck Workshop, February 21-22, 2008

  4. Preference model • To solve a multicriteria decision problem one needs a preference model, i.e. criteria aggregation model • Traditional aggregation paradigm: The preference model is first constructed and then applied on set A to get information about the comprehensive preference • Disaggregation-aggregation (ordinal regression) paradigm: The comprehensive preference on a subset AR A is known a priori, and a consistent preference model is inferred from this information to be applied on set A 2nd Decision Deck Workshop, February 21-22, 2008

  5. Disaggregation-aggregation (regression) approach • The additive utility function is defined on A as follows: U(x) = Σui(xi), i I = {1, …, m} The preference model is a set of additive utility functions compatible with a non-complete set of pairwise comparisons of some reference actions and information about comprehensive and partial intensities of preference 2nd Decision Deck Workshop, February 21-22, 2008

  6. The UTAGMS method (Greco, Mousseau & Słowiński 2004) • BR ARxAR is the set of pairs of reference actions compared by the DM • The preference information is a partial preorder  on a subset of reference actions ARA •  – weak preference (outranking) relation for each pair (x, y) BR xy „x is at least as good as y” xy [xy and not yx]  „x is preferred to y” x~y [xy and yx]  „x is indifferent to y” • A utility function is called compatible if it is able to restore all pairwise comparisons from BR (i.e. partial preorder) on AR 2nd Decision Deck Workshop, February 21-22, 2008

  7. preference information A BR xy zw yv ut zu uz All instances ofpreference model compatible with preference information DM analyst y x AR t z v w u Apply all compatible instances on A The UTAGMS method (Greco, Mousseau & Słowiński 2004) • Questions: • Are any two actions x, yA ordered in the same way by all compatible utility functions? • Is there at least one compatible utility function ordering x at least as good as y (or y at least as good as x)? 2nd Decision Deck Workshop, February 21-22, 2008

  8. The UTAGMS method (Greco, Mousseau & Słowiński 2004) • Having answers to these questions for all pair of actions (x, y) A x A, one gets: • necessary weak preferencerelationN , whose semantics is U(x)  U(y) for all compatible utility functions • possible weakpreference relationP, whosesemantics isU(x)  U(y) for at least one compatible utility function • The necessary and possible weak preference relations are exploited such that one finally obtains two rankings in the set of actions: • necessary ranking (partial preorder) • possible ranking (complete and negatively transitive binary relation) 2nd Decision Deck Workshop, February 21-22, 2008

  9. xy zw yv ut zu uz preference information The UTAGMS method (Greco, Mousseau & Słowiński 2004) • Two rankings result: necessary and possible Includesnecessary ranking and does not include the complement of necessary ranking z u x w y t v possible ranking necessary ranking 2nd Decision Deck Workshop, February 21-22, 2008

  10. The UTAGMS method (Greco, Mousseau & Słowiński 2004) • For any pair of actions (x,y)A, and for available preference information represented by BR, preference of x over y is determined by compatible utility functions U verifying set E(x,y) of constraints: U’(x) U’(y) +   x  y U’(x) = U’(y)  x ~ y ui(xij) – ui(xij-1)  0, i = 1, …, m, j = 1, …, ω + 1 ui(xi0) = 0, i = 1, …, m Σui(xiω+1) = 1, i = 1, …, m where  is a small positive constant, and ω = m + 2 - |AR  {x, y}| (x, y)  BR E(x, y) 2nd Decision Deck Workshop, February 21-22, 2008

  11. The UTAGMS method (Greco, Mousseau & Słowiński 2004) • Given a pair of actions x,yA xNy d(x,y)  0 where d(x, y) = Min {U(x) – U(y)} s.t. E(x, y) • d(x,y)  0 means that for allcompatible utility functionsx is at least as good as y • For any (x,y)BR : xy xNy 2nd Decision Deck Workshop, February 21-22, 2008

  12. The UTAGMS method (Greco, Mousseau & Słowiński 2004) • Given a pair of actions x,yA xPy D(x,y)  0 where D(x, y) = Max {U(x) – U(y)} s.t. E(x, y) • d(x,y)  0 means that for at least onecompatible utilityfunctions x is at least as good as y • For any (x,y)BR : xy xP y 2nd Decision Deck Workshop, February 21-22, 2008

  13. The GRIP method (Figueira, Greco & Słowiński 2006) • GRIP (Generalized Regression with Intensities of Preference) extends UTAGMS method by adopting all features of UTAGMSand by taking into account additional preference information: • comprehensive comparisons of intensities of preference between some pairs of reference actions, e.g. „x is preferred to y at least as much as w is preferred to z” • partial comparisons of intensities of preference between some pairs of reference actions on particular criteria,e.g. „x is preferred to y at least as much as w is preferred to z,on criterion giF” 2nd Decision Deck Workshop, February 21-22, 2008

  14. The GRIP method (Figueira, Greco & Słowiński 2006) • DM is supposed to provide the following preference information: • a partial preorder on AR, such that x,yAR xy  „x is at least as good as y”  =  non-1,  = -1 • a partial preorder * on ARAR, such that x,y,w,zAR (x,y) * (w,z) „x is preferred to y at least as much as w is preferred to z” * = *  non*-1, * = *  *-1 • a partial preorder i* on ARAR, i =1,...,m, such that x,y,w,zAR (x,y) i* (w,z) „x is preferred to y at least as much as w is preferred to z,on criterion giF” i* = i*  noni*-1, i* = i*  i*-1 2nd Decision Deck Workshop, February 21-22, 2008

  15. The GRIP method (Figueira, Greco & Słowiński 2006) • A utility function Uis called compatible if it satisfies the constraints corresponding to DM’s preference information: • U(x)  U(y) iff xy • U(x) > U(y) iff x y • U(x) = U(y) iff x y • U(x) – U(y)  U(w) – U(z) iff (x,y) * (w,z) • U(x) – U(y) > U(w) – U(z) iff (x,y) * (w,z) • U(x) – U(y) = U(w) – U(z) iff (x,y) * (w,z) • ui(x)  ui(y) iff xiy, iI • ui(x) – ui(y)  ui(w) – ui(z) iff (x,y) i* (w,z), iI • ui(x) – ui(y) > ui(w) – ui(z) iff (x,y) i* (w,z), iI • ui(x) – ui(y) = ui(w) – ui(z) iff (x,y) i* (w,z), iI 2nd Decision Deck Workshop, February 21-22, 2008

  16. The GRIP method (Figueira, Greco & Słowiński 2006) • Moreover, the following normalization constraints should also be taken into account: • ui(xi*) = 0, i  I where xi* is such that xi* = min {gi(x): xA} • Σ ui(yi*) = 1, i I where yi* is such that yi* = max {gi(y): xA} • Let as remark that like in UTAGMS method, constraints b), e) and i) should be written as: b’)U(x)  U(y) +  e’)U(x) – U(y)  U(w) – U(z) +  i’)ui(x) – ui(y)  ui(w) – ui(z) +  where  is a small positive constant 2nd Decision Deck Workshop, February 21-22, 2008

  17. The GRIP method (Figueira, Greco & Słowiński 2006) • If constraints a) – l)are consistent, then we gettwo weak preference relationsN and P , and two binary relations comparing intensity of preference*Nand *P: • for all x,yA, a necessary weak preference relation xNy min {U(x) – U(y)}  0 • for all x,yA, a possible weak preference relation xPy max {U(x) – U(y)}  0 • for all x,y,w,z A, a necessary relation of preference intensity (x,y) *N(w,z) min {[U(x) – U(y)] – [U(w) – U(z)]}  0 • for all x,y,w,z A, a possible relation of preference intensity (x,y) *P(w,z) max {[U(x) – U(y)] – [U(w) – U(z)]}  0 where „min” and „max” are calculated over all utility functions satisfying a) – l) 2nd Decision Deck Workshop, February 21-22, 2008

  18. The GRIP method (Figueira, Greco & Słowiński 2006) • In order to conclude the truth or falsity of necessary and possible weak preference relations N, P and *N, *P, one can use LP • To obtain the result which is independent on the value of , one should: Max   subject to constraints a)–l), with b),e),i) written as b’),e’),i’) • If maximal *>0, the set of compatible utility functions is not empty 2nd Decision Deck Workshop, February 21-22, 2008

  19. The GRIP method (Figueira, Greco & Słowiński 2006) • Then, to verify the truth or falsity of xPy,for anyx,yA, one should: Max   subject to constraints a)–l), with b),e),i) written as b’),e’),i’) and U(x)  U(y) • Maximal *>0  xPy This means that there exists at least one compatible utility function satisfying the hypothesis U(x)  U(y) 2nd Decision Deck Workshop, February 21-22, 2008

  20. The GRIP method (Figueira, Greco & Słowiński 2006) • In order to verify the truth or falsity of x N y,rather than to check directly that for each compatible utility function U(x)  U(y), we make sure that among the compatible utility functions there is no one such that U(x) < U(y): Max   subject to constraints a) – l), with b), e), i) written as b’), e’), i’) and U(y)  U(x) +  • Maximal * ≤ 0  x Ny 2nd Decision Deck Workshop, February 21-22, 2008

  21. The GRIP method (Figueira, Greco & Słowiński 2006) • Analogously, in order to verify the truth or falsity of (x,y)*P(w,z)for anyx,y,w,zA, one should: Max   subject to constraints a)–l), with b),e),i) written as b’),e’),i’) and U(x)U(y)  U(w)U(z) • Maximal *>0  (x,y)*P(w,z) 2nd Decision Deck Workshop, February 21-22, 2008

  22. The GRIP method (Figueira, Greco & Słowiński 2006) • Analogously, in order to verify the truth or falsity of (x,y)*N (w,z)for anyx,y,w,zA, one should: Max   subject to constraints a)–l), with b),e),i) written as b’),e’),i’) and U(w)U(z)  U(x)U(y) +  • Maximal *≤0  (x,y)*P(w,z) • The value of*is not meaningful – the result does not depend on it 2nd Decision Deck Workshop, February 21-22, 2008

  23. UTAGMS/GRIP plugin overview • Current implementation of UTAGMS/GRIP plugin works on the first version of Decision Deck platform (1.0.2) • To verify the truth or falsity of preference relations it uses GLKP linear solver which is the part of D2 platform (GLPK plugin) • To visualize rankings of alternatives in the form of graph it uses the JGraph library implemented as additional plugin • UTAGMS/GRIP plugin main features: • add/remove alternatives to/from reference set • add/remove/edit preference information (partial preorder, comprehensive and/or partial intensities of preferences) • shows comparison of alternatives • view necessary ranking of alternatives 2nd Decision Deck Workshop, February 21-22, 2008

  24. UTAGMS/GRIP plugin demonstration • Illustrative example Car ranking problem 2nd Decision Deck Workshop, February 21-22, 2008

  25. UTAGMS/GRIP plugin demonstration • Illustrative example – Car ranking problem Alternatives: Criteria: 2nd Decision Deck Workshop, February 21-22, 2008

  26. Price Speed Space Fuel_cons. Acceleration Skoda Opel Ford Citroen Seat VW UTAGMS/GRIP plugin demonstration Performance matrix: 2nd Decision Deck Workshop, February 21-22, 2008

  27. Conclusions and future works • The preference information used in GRIP does not need to be complete: the DM can compare only those pairs of reference alternatives on particular criteria for which his/her judgment is sufficiently certain • Distinguishing necessary and possible consequences of preference information, GRIP answers questions of robustness analysis using all utility functions instead of a single „best-fit” utility function • Plugin future works: • visualization of possible ranking of alternatives • resolving inconsistency in preference information • visualization of necessary and possible relations of preference intensity for the pair of alternatives • manage preference information using „classes of attractiveness” 2nd Decision Deck Workshop, February 21-22, 2008

More Related