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

2. 2nd Decision Deck Workshop, February 21-22, 20082 Plan Problem statement Problem statement Disaggregation-aggregation (regression) approach Disaggregation-aggregation (regression) approach UTA GMS method UTA GMS method GRIP method GRIP method UTA GMS /GRIP plugin overview UTA GMS /GRIP plugin overview UTA GMS /GRIP plugin demonstration UTA GMS /GRIP plugin demonstration Conclusions and future works

3. 2nd Decision Deck Workshop, February 21-22, 20083 Problem statement Consider a finite set A of alternatives (actions) evaluated by m criteria from a consistent family F = {g 1,...,g m } Consider a finite set A of alternatives (actions) evaluated by m criteria from a consistent family F = {g 1,...,g m } Taking into account preferences of a Decision Maker (DM), rank all the actions of set A from the best to the worst Taking into account preferences of a Decision Maker (DM), rank all the actions of set A from the best to the worst A * * * * x x x x x x * * x x * x * x x x * x x x x x

4. 2nd Decision Deck Workshop, February 21-22, 20084 Preference model To solve a multicriteria decision problem one needs a preference model, i.e. criteria aggregation model To solve a multicriteria decision problem one needs a preference model, i.e. criteria aggregation model Traditional aggregation paradigm: 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: Disaggregation-aggregation (ordinal regression) paradigm: The comprehensive preference on a subset A R  A is known a priori, and a consistent preference model is inferred from this information to be applied on set A

5. 2nd Decision Deck Workshop, February 21-22, 20085 Disaggregation-aggregation (regression) approach 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 The additive utility function is defined on A as follows: The additive utility function is defined on A as follows: U(x) = Σ u i (x i ), i  I = {1, …, m}

6. 2nd Decision Deck Workshop, February 21-22, 20086 The UTA GMS method (Greco, Mousseau & Słowiński 2004) B R  A R x A R is the set of pairs of reference actions compared by the DM B R  A R x A R 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 A R  A The preference information is a partial preorder  on a subset of reference actions A R  A  – weak preference (outranking) relation  – weak preference (outranking) relation for each pair (x, y)  B R 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 B R (i.e. partial preorder) on A R A utility function is called compatible if it is able to restore all pairwise comparisons from B R (i.e. partial preorder) on A R

7. 2nd Decision Deck Workshop, February 21-22, 20087 The UTA GMS method (Greco, Mousseau & Słowiński 2004) Questions: 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)? A ARAR x tz w v y u DM x  yz  wy  vu  tz  uu  zx  yz  wy  vu  tz  uu  z preference information analyst All instances of preference model compatible with preference information BRBR Apply all compatible instances on A

8. 2nd Decision Deck Workshop, February 21-22, 20088 The UTA GMS method (Greco, Mousseau & Słowiński 2004) Having answers to these questions for all pair of actions (x, y)  A x A, one gets: Having answers to these questions for all pair of actions (x, y)  A x A, one gets:  necessary weak preference relation U(x)  U(y) for all compatible utility functions  necessary weak preference relation  N, whose semantics is U(x)  U(y) for all compatible utility functions  possible weak preference relation, whose semantics is U(x)  U(y) for at least one compatible utility function  possible weak preference relation  P, whose semantics is U(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: 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)

9. 2nd Decision Deck Workshop, February 21-22, 20089 The UTA GMS method (Greco, Mousseau & Słowiński 2004) Two rankings result: necessary and possible Two rankings result: necessary and possible x y w z t u v necessary ranking possible ranking Includes necessary ranking and does not include the complement of necessary ranking x  yz  wy  vu  tz  uu  zx  yz  wy  vu  tz  uu  z preference information

10. 2nd Decision Deck Workshop, February 21-22, 200810 The UTA GMS method (Greco, Mousseau & Słowiński 2004) For any pair of actions (x, y)  A, and for available preference information represented by B R, preference of x over y is determined by compatible utility functions U verifying set E(x, y) of constraints: For any pair of actions (x, y)  A, and for available preference information represented by B R, 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 u i (x i j ) – u i (x i j-1 )  0, i = 1, …, m, j = 1, …, ω + 1 u i (x i 0 ) = 0, i = 1, …, m Σ u i (x i ω+1 ) = 1, i = 1, …, m where  is a small positive constant, and ω = m + 2 - |A R  {x, y}|  (x, y)  B R E(x, y)

11. 2nd Decision Deck Workshop, February 21-22, 200811 The UTA GMS method (Greco, Mousseau & Słowiński 2004) Given a pair of actions x, y  A Given a pair of actions x, y  A x  N y  d(x, y)  0 x  N y  d(x, y)  0 where where d(x, y) = Min {U(x) – U(y)} d(x, y) = Min {U(x) – U(y)} s.t. E(x, y) s.t. E(x, y) d(x, y)  0 means that for all compatible utility functions x is at least as good as y d(x, y)  0 means that for all compatible utility functions x is at least as good as y For any (x, y)  B R : For any (x, y)  B R : x  y  x  N y x  y  x  N y

12. 2nd Decision Deck Workshop, February 21-22, 200812 The UTA GMS method (Greco, Mousseau & Słowiński 2004) Given a pair of actions x, y  A Given a pair of actions x, y  A x  P y  D(x, y)  0 x  P y  D(x, y)  0 where where D(x, y) = Max {U(x) – U(y)} D(x, y) = Max {U(x) – U(y)} s.t. E(x, y) s.t. E(x, y) d(x, y)  0 means that for at least one compatible utility functions x is at least as good as y d(x, y)  0 means that for at least one compatible utility functions x is at least as good as y For any (x, y)  B R : For any (x, y)  B R : x  y  x  P y x  y  x  P y

13. 2nd Decision Deck Workshop, February 21-22, 200813 The GRIP method (Figueira, Greco & Słowiński 2006) GRIP (Generalized Regression with Intensities of Preference) extends UTA GMS method by adopting all features of UTA GMS and by taking into account additional preference information: GRIP (Generalized Regression with Intensities of Preference) extends UTA GMS method by adopting all features of UTA GMS and 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 g i  F”

14. 2nd Decision Deck Workshop, February 21-22, 200814 The GRIP method (Figueira, Greco & Słowiński 2006) DM is supposed to provide the following preference information: DM is supposed to provide the following preference information:  a partial preorder  on A R, such that  x, y  A R x  y  „x is at least as good as y” x  y  „x is at least as good as y”  =   non  -1,  =    -1  =   non  -1,  =    -1  a partial preorder  * on A R  A R, such that  x, y, w, z  A R (x, y)  * (w, z)  „x is preferred to y at least as much as w is (x, y)  * (w, z)  „x is preferred to y at least as much as w is preferred to z”  * =  *  non  * -1,  * =  *   * -1 preferred to z”  * =  *  non  * -1,  * =  *   * -1  a partial preorder  i * on A R  A R, i = 1,..., m, such that  x, y, w, z  A R (x, y)  i * (w, z)  „x is preferred to y at least as much as w is (x, y)  i * (w, z)  „x is preferred to y at least as much as w is preferred to z, on criterion g i  F” preferred to z, on criterion g i  F”  i * =  i *  non  i * -1,  i * =  i *   i * -1  i * =  i *  non  i * -1,  i * =  i *   i * -1

15. 2nd Decision Deck Workshop, February 21-22, 200815 The GRIP method (Figueira, Greco & Słowiński 2006) A utility function U is called compatible if it satisfies the constraints corresponding to DM’s preference information: A utility function U is called compatible if it satisfies the constraints corresponding to DM’s preference information: a) U(x)  U(y) iff x  y b) U(x) > U(y) iff x  y c) U(x) = U(y) iff x  y d) U(x) – U(y)  U(w) – U(z) iff (x, y)  * (w, z) e) U(x) – U(y) > U(w) – U(z) iff (x, y)  * (w, z) f) U(x) – U(y) = U(w) – U(z) iff (x, y)  * (w, z) g) u i (x)  u i (y) iff x  i y, i  I h) u i (x) – u i (y)  u i (w) – u i (z) iff (x, y)  i * (w, z), i  I i) u i (x) – u i (y) > u i (w) – u i (z) iff (x, y)  i * (w, z), i  I j) u i (x) – u i (y) = u i (w) – u i (z) iff (x, y)  i * (w, z), i  I

16. 2nd Decision Deck Workshop, February 21-22, 200816 The GRIP method (Figueira, Greco & Słowiński 2006) Moreover, the following normalization constraints should also be taken into account: Moreover, the following normalization constraints should also be taken into account: k) u i (x i * ) = 0, i  I where x i * is such that x i * = min {g i (x): x A} where x i * is such that x i * = min {g i (x): x  A} l) Σ u i (y i * ) = 1, i  I where y i * is such that y i * = max {g i (y): x A} where y i * is such that y i * = max {g i (y): x  A} Let as remark that like in UTA GMS method, constraints b), e) and i) should be written as: Let as remark that like in UTA GMS 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’) u i (x) – u i (y)  u i (w) – u i (z) +  where  is a small positive constant

17. 2nd Decision Deck Workshop, February 21-22, 200817 The GRIP method (Figueira, Greco & Słowiński 2006) If constraints a) – l) are consistent, then we get two weak preference relations  N and  P, and two binary relations comparing intensity of preference  * N and  * P : If constraints a) – l) are consistent, then we get two weak preference relations  N and  P, and two binary relations comparing intensity of preference  * N and  * P : 1.for all x, y  A, a necessary weak preference relation x  N y  min {U(x) – U(y)}  0 x  N y  min {U(x) – U(y)}  0 2.for all x, y  A, a possible weak preference relation x  P y  max {U(x) – U(y)}  0 x  P y  max {U(x) – U(y)}  0 3.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 (x, y)  * N (w, z)  min {[U(x) – U(y)] – [U(w) – U(z)]}  0 4.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 (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)

18. 2nd Decision Deck Workshop, February 21-22, 200818 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 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: 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 If maximal  * > 0, the set of compatible utility functions is not empty

19. 2nd Decision Deck Workshop, February 21-22, 200819 The GRIP method (Figueira, Greco & Słowiński 2006) Then, to verify the truth or falsity of x  P y, for any x, y  A, one should: Then, to verify the truth or falsity of x  P y, for any x, y  A, one should: Max   subject to constraints a) – l), with b), e), i) written as b’), e’), i’) b’), e’), i’) and U(x)  U(y) Maximal  * > 0  x  P y Maximal  * > 0  x  P y This means that there exists at least one compatible utility function satisfying the hypothesis U(x)  U(y)

20. 2nd Decision Deck Workshop, February 21-22, 200820 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): 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’) b’), e’), i’) and U(y)  U(x) +  Maximal  * ≤ 0  x  N y Maximal  * ≤ 0  x  N y

21. 2nd Decision Deck Workshop, February 21-22, 200821 The GRIP method (Figueira, Greco & Słowiński 2006) Analogously, in order to verify the truth or falsity of (x, y)  * P (w, z) for any x, y, w, z  A, one should: Analogously, in order to verify the truth or falsity of (x, y)  * P (w, z) for any x, 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) Maximal  * > 0  (x, y)  * P (w, z)

22. 2nd Decision Deck Workshop, February 21-22, 200822 The GRIP method (Figueira, Greco & Słowiński 2006) Analogously, in order to verify the truth or falsity of (x, y)  * N (w, z) for any x, y, w, z  A, one should: Analogously, in order to verify the truth or falsity of (x, y)  * N (w, z) for any x, 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) Maximal  * ≤ 0  (x, y)  * P (w, z) The value of  * is not meaningful – the result does not depend on it The value of  * is not meaningful – the result does not depend on it

23. 2nd Decision Deck Workshop, February 21-22, 200823 UTA GMS /GRIP plugin overview Current implementation of UTA GMS /GRIP plugin works on the first version of Decision Deck platform (1.0.2) Current implementation of UTA GMS /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 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 To visualize rankings of alternatives in the form of graph it uses the JGraph library implemented as additional plugin UTA GMS /GRIP plugin main features: UTA GMS /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

24. 2nd Decision Deck Workshop, February 21-22, 200824 UTA GMS /GRIP plugin demonstration Illustrative example Illustrative example Car ranking problem

25. 2nd Decision Deck Workshop, February 21-22, 200825 UTA GMS /GRIP plugin demonstration Illustrative example – Car ranking problem Illustrative example – Car ranking problemAlternatives:Criteria:

26. 2nd Decision Deck Workshop, February 21-22, 200826 UTA GMS /GRIP plugin demonstration Performance matrix: Skoda Opel Ford Citroen Seat VW Price Speed Space Fuel_cons. Acceleration

27. 2nd Decision Deck Workshop, February 21-22, 200827 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 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 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: 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”