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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

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to the Decision Deck platform

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  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

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