Roman Słowiński Poznań University of Technology, Poland

1. Multiple-criteria ranking using an additive value function constructed via ordinal regresion :UTA method Roman Słowiński Poznań University of Technology, Poland  Roman Słowiński

2. g2(x) g2max A g2min g1(x) g1min g1max Problem statement • Consider a finite set A of actions (actions, solutions, objects) evaluated by m criteria from a consistent family F={g1,...,gm} • Let I={1,…,m}

3. What is a consistent family of criteria ? • A family of criteria F={g1,...,gn} is consistent if it is: • Complete – if two actions have the same evaluations on all criteria, then they have to be indifferent, i.e. if for any a,bA, there is gi(a)~gi(b), iI, then a~b • Monotonic – if action a is preferred to action b (ab), and there is action c, such that gi(c)gi(a), iI, then cb • Non-redundant – elimination of any criterion from the family F should violate at least one of the above properties

4. A x * * x * * * x x * * x x x * * * x x x x x x * * x x x x Problem statement • Taking into account preferences of a Decision Maker (DM), rank all the actions of set A from the best to the worst x

5. g2(x) g2max nadir A ideal g2min g1(x) g1min g1max Dominance relation • Action aA is non-dominated (Pareto-optimal) if and only if there is no other action bA such that gi(b)gi(a), iI, and on at least one criterion jI, gi(b)gi(a)

6. Criteria aggregation model = preference model • Dominance relation is too poor – it leaves many actions non-comparable • One can „enrich” the dominance relation, using preference information elicited from the Decision Maker • Preference information permits to built a preference model that aggregates the vector evaluations of elements of A

7. Why traditional MCDM methods may confuse their users ? • Traditional MCDM methods require a rich and difficult preference information: • many intracriteria and intercriteria parameters: thresholds, weights, … • complete set of pairwise comparisons of actions on each criterion • complete set of pairwise comparisons of criteria • … • They suppose the DM understands the logic of a particular aggregation model: • meaning of weights: substitution ratios or relative strengths • meaning of lotteries (ASSESS) • meaning of indifference, preference and veto thresholds (ELECTRE) • meaning of the ratio scale of the intensity of preference (AHP) • meaning of „neutral” and „good” levels on particular criteria (MACBETH) • …

8. Towards „easy” preference information • Traditional methods appear to be too demanding of cognitive effort of their users • This is why we advocate for methods requiring „easy” preference information • „Easy” means natural and even partial • Psychologists confirm that DMs are more confident exercising their decisions than explaining them

9. A Towards „easy” preference information • The most natural is a holistic pairwise comparison of some actions relatively well known to the DM, i.e. reference actions

10. holistic preference information A xy zw xw yv ut zu uz DM y x AR t z v w u Towards „easy” preference information • The most natural is a holistic pairwise comparison of some actions relatively well known to the DM, i.e. reference actions

11. preference information A xy zw xw yv ut zu uz Preference model compatible with preference information DM analyst y x AR t z v w u Apply the preference model on A Towards „easy” preference information • Question: what is the consequence of using on the whole set Athis information transformed to a compatible preference model ? What ranking will result ?

12. Aggregation paradigms • Disaggregation-aggregation (or regression) paradigm: The holistic preference on a subset ARAis known first, and then a compatible criteria aggregation model (compatible preference model)is inferred from this information to be applied on set A • Traditional aggregation paradigm: The criteria aggregation model (preference model) is first constructed and then applied on set A to get information about holistic preference

13. Aggregation paradigms • The disaggregation-aggregation paradigam has been introduced to MCDS by Jacquet-Lagreze & Siskos(1982) in the UTA method: the inferred criteria aggregation model is the additive value function with piecewise-linear marginal value functions • The disaggregation-aggregation paradigam is consistent with the „posterior rationality” principle by March(1988) and „learning from examples” used in AI and knowledge discovery • Other aggregation models inferred in this way: • Fishburn (1967) – trade-off weights • Mousseau & Słowiński(1998) – outranking relation (ELECTRE TRI) • Greco, Matarazzo & Słowiński(1999) – decision rules or trees (DRSA – Dominance-based Rough Set Approach)

14. Basic concepts and notation • Gi – domain of criterion gi (Gi is finite or countably infinte) • – evaluation space • x,yG – profiles of actions in evaluation space • – weak preference(outranking) relation onG:for each x,yG xy„x is at least as good as y” xy [xyand notyx] „x is preferred to y” x~y [xyand yx] „x is indifferent to y”

15. Reminder of the UTA method (Jacquet-Lagreze & Siskos, 1982) • For simplicity: Gi, iI, where I={1,…,m} • For eachgi,Gi=[i, i] is the criterion evaluation scale, i  i , where i and i, are the worst and the best (finite) evaluations, resp. • Thus, A is a finite subset of G and • Additive value(or utility) function on G: for each xG where ui are non-decreasing marginal value functions, ui : Gi, iI

16. x1 A  x2  AR x3x4  x5  x6x7 Principle of the ordinal regression - UTA (Jacquet-Lagreze & Siskos, 1982) • The preference information is given in form of a complete preorder on a subset of reference actionsARA, AR={x1,x2,...,xn} – the reference actions are rearranged such that xkxk+1 , k=1,...,n-1

17. a1  a2  a3 Principle of the ordinal regression • Example: Let AR={a1, a2, a3}, G={Gain_1, Gain_2} Evaluation of reference actions on criteria Gain_1, Gain_2: Reference ranking:

18. Principle of the ordinal regression

19. Principle of the ordinal regression

20. a1 a1   a3 a2   a2 a3 Principle of the ordinal regression • Let’s change the reference ranking: • One linear piece per each marginal value function u1, u2 is not enough u1=k1Gain_1, u2=k2Gain_2, U=u1+u2 For a1a3, k2>k1, but for a3a2, k1>k2, thus, marginal value functions cannot be linear

21. Principle of the ordinal regression

22. Principle of the UTA method (Jacquet-Lagreze & Siskos, 1982) • The comprehensive preference information is given in form of a complete preorder on a subset of reference actionsARA, AR={x1,x2,...,xn} – the reference actions are rearranged such that xkxk+1 , k=1,...,n-1 • The inferred value of each reference action xAR where + and - are potential errors of over- and under-estimation of the right value, resp. • The intervals [i, i] are divided into iequal sub-intervals with the end points (iI)

23. Principle of the ordinal regression • In the UTA method, the marginal value of action xA is approximated by linear interpolation: for

24. UTA additive preference model

25. Principle of the UTA method • Ordinal regression principle for xkxk+1 , k=1,...,n-1 • Monotonicity of preferences • Normalization

26. k=1,...,n-1 (C) Principle of the UTA method • The marginal value functions (breakpoint variables) are estimated by solving the LP problem where  is a small positive constant

27. polyhedron of constraints (C) EUTA EUTA*+ EUTA= EUTA* Principle of the UTA method • If EUTA*=0, then the polyhedron of feasible solutions for ui(xi) is not empty and there exists at least one value functionU[g(x)] compatible with the complete preorder on AR • If EUTA*>0, then there is no value functionU[g(x)] compatible with the complete preorder on AR– three possible moves: • increasing the number of linear pieces i for ui(xi) • revision of the complete preorder on AR • post-optimal search for the best function with respect to Kendall’s  in the area EUTA EUTA*+  Jacquet-Lagreze & Siskos (1982)

28. Współczynnik Kendalla • Do wyznaczania odległości między preporządkami stosuje się miarę Kendalla • Przyjmijmy, że mamy dwie macierze kwadratowe R i R* o rozmiarze m m, gdzie m = |AR|, czyli m jest liczbą wariantów referencyjnych • macierz R jest związana z porządkiem referencyjnym podanym przez decydenta, • macierz R* jest związana z porządkiem dokonanym przez funkcję użyteczności wyznaczoną z zadania PL (zadania regresji porządkowej) • Każdy element macierzy R, czyli rij (i, j=1,..,m), może przyjmować wartości: • To samo dotyczy elementów macierzy R* • Tak więc w każdej z tych macierzy kodujemy pozycję (w porządku) wariantu a względem wariantu b

29. Współczynnik Kendalla • Następnie oblicza się współczynnik Kendalla: gdzie dk(R,R*) jest odległością Kendallamiędzy macierzami R i R*: • Stąd -1, 1 • Jeżeli  = -1, to oznacza to, że porządki zakodowane w macierzach R i R*są zupełnie odwrotne, np. macierz R koduje porządek a  b  c  d, a macierz R* porządek d  c  b  a • Jeżeli  = 1, to zachodzi całkowita zgodność porządków z obydwu macierzy. W tej sytuacji błąd estymacji funkcji użyteczności F*=0 • W praktyce funkcję użyteczności akceptuje się, gdy  0.75

30. 1 2 Example of UTA+ • Ranking of 6 means of transportation

31. Preference attitude: „economical”

32. Preference attitude: „hurry”

33. Preference attitude: „hurry”

34. preference information xy zw yv ut zu uz All instances ofpreference model compatible with preference information A DM analyst y x AR t z v w u Apply all compatible instances on A One should use all compatible preference models on set A • Question: what is the consequence of using all compatible preference models on set A ? What rankings will result ?

35. z u Includesnecessary ranking and does not include the complement of necessary ranking x w xy zw yv ut zu uz preference information y t v possible ranking necessary ranking Two rankings result: necessary and possible

36. z u Includes necessary ranking and does not include the complement of necessary ranking w x xy zw yv ut zu uz xw additional preference information y t v necessary rankingenriched possible rankingimpoverished Two rankings result: necessary and possible – effect of additional preference information „trial-and-error” interactions • In the absence of any preference information: • necessary ranking boils down to weak dominance relation • possible ranking is a complete relation • For complete pairwise comparisons (complete preorder in A): • necessary ranking = possible ranking

37. The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The preference information is a partial preorder on a subset of reference actions ARA • A value function is called compatible if it is able to restore the partial preorder reference actions from AR • Each compatible value function induces a ranking on set A • In result, one obtains two rankings on set A, such that for any pair of actions (x,y)A: • x N y:x is ranked at least as good as y iff U(x)U(y) for all value functions compatible withthe preference information (necessary weak preference relation N - a partial preorder on A) • x P y:x is ranked at least as good as y iff U(x)U(y) for at least one value function compatible with the preference information (possibleweak preference relation P - a strongly complete and negatively transitive binary relation on A)

38. ui(xi) gi 0 i i yi vi wi zi The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The marginal value function ui(xi) y,v,w,zAR Characteristic points of marginal value functions are fixed on actual evaluations of actions from set A

39. ui(xi) ? ? ? ? ? gi 0 i i yi vi wi zi The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The marginal value function ui(xi) y,v,w,zAR Marginal values in characteristic points are unknown

40. ui(xi) gi 0 i i yi vi wi zi The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The marginal value function ui(xi) y,v,w,zAR In fact, they are intervals, because all compatible value functions are considered

41. ui(xi) gi 0 i i yi vi wi zi The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The marginal value function ui(xi) y,v,w,zAR The area of all compatible marginal value functions

42. ui(xi) gi 0 i i yi vi wi zi The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The marginal value function ui(xi) y,v,w,zAR In the area the marginal compatible value functions must be monotone

43. ui(xi) gi 0 i i yi vi wi zi The UTAGMS method(Greco, Mousseau & Słowiński 2004) • The marginal value function ui(xi) • This means that the ordinal regression should not seek for m piecewise-linear marginal value functions, but for any compatible additive value function

44. The UTAGMS method • Let ibe a permutation on the set of actions AR{x,y} that reorders them according to increasing evaluation on criterion gi: where • if AR{x,y}=, then =n+2 • if AR{x,y}={x} or AR{x,y}={y}, then =n+1 • if AR{x,y}={x,y}, then =n • The characteristic points of ui(xi), iI, are then fixen in:

45. a a b b  a,bAR a a b b E(x,y) a a a The UTAGMS method • For any pair of actions (x,y)A, and for available preference information concerning AR, preference of x over y is determined by compatible value functionsU verifying set E(x,y) of constraints: where  is a small positive constant • For all (x,y)A, E(x,y) = E(y,x)

46. The UTAGMS method • Nmeans necessary(strong) preference relation • Given a pair of actions x,yA xNyd(x,y)  0 where • d(x,y)  0 means that for allcompatible value functions x is at least as good as y • For x,yAR : xy xNy

47. The UTAGMS method • Pmeans possible(weak) preference relation • Given a pair of actions x,yA xPyD(x,y)  0 where • D(x,y)  0 means that for at least onecompatible value function x is at least as good as y • For x,yAR : xy notyPx

48. The UTAGMS method • Some properties: • xNyxPy • Nis a partial preorder (i.e. Nis reflexive and transitive) • Pis strongly complete (i.e. for all x,yA, xPyor yPx)and negatively transitive (i.e. for all x,y,zA, notxPyand not yPznot xPz), (in general, Pis not transitive) • d(x,y) = Min{U(x)–U(y)} = –Max{–[U(x)–U(y)]} = = –Max{U(y)–U(x)} = –D(y,x)

49. Proof of transitivity of N • d(x,y)>0 means: Min{U(x)-U(y)}>0 • This is equivalent to the fact: for all value functions compatible with the reference preorder, U(x)>U(y) • The set of all compatible value functions is the same for calculation of d(x,y) for any pair x,yA • Suppose, the transitivity of N is not true, i.e. for x,y,zA Min{U(x)-U(y)}>0, Min{U(y)-U(z)}>0, but Min{U(x)-U(z)}<0 • This means that U(x)-U(z) has achieved a minimum value d(x,z)<0 for a value function denoted by U*, such that U*(x)<U*(z), while U*(x)>U*(y) and U*(y)>U*(z) • In other words, U*(x)>U*(y)>U*(z)>U*(x) • This is a contradiction, so N is transitive

50. The UTAGMS method • Elaboration of the final ranking: • for the necessary preference relation being a partial preorder (N is supported by all compatible value functions) preference: xNy if xNy and not yNx indifference: xNy if xNy and yNx incomparability: x ? y if notxNy and not yNx • for the possible preference relation being complete (P is supported by at least one compatible value function) preference: xPy if xPy and not yPx indifference: xPy if xPy and yPx • N.B. It is impossible to infer one ranking from another because strong and weak outranking relations arenot dual