Multiplecriteria ranking using an additive value function constructed

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

Problem statement g 2(x) g 2 max n Consider a finite set A of actions (actions, solutions, objects) A evaluated by m criteria from a consistent family F={g 1, . . . , gm} n g Let I={1, …, m} 2 min g 1(x) g 1 min g 1 max 2

What is a consistent family of criteria ? n A family of criteria F={g 1, . . . , gn} is consistent if it is: n 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 n 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 n Non-redundant – elimination of any criterion from the family F should violate at least one of the above properties 3

Problem statement n 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 4

Dominance relation n 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) g 2(x) g 2 max nadir A g 2 min ideal g 1(x) g 1 min g 1 max 5

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

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

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

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

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

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

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

Aggregation paradigms n 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 n The disaggregation-aggregation paradigam is consistent with the „posterior rationality” principle by March (1988) and „learning from examples” used in AI and knowledge discovery n Other aggregation models inferred in this way: n Fishburn (1967) – trade-off weights n Mousseau & Słowiński (1998) – outranking relation (ELECTRE TRI) n Greco, Matarazzo & Słowiński (1999) – decision rules or trees (DRSA – Dominance-based Rough Set Approach) 13

Basic concepts and notation n n Gi – domain of criterion gi (Gi is finite or countably infinte) – evaluation space n x, y G – profiles of actions in evaluation space n – weak preference (outranking) relation on G: for each x, y G 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” 14

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

Principle of the ordinal regression - UTA n (Jacquet-Lagreze & Siskos, 1982) The preference information is given in form of a complete preorder on a subset of reference actions AR A, AR={x 1, x 2, . . . , xn} – the reference actions are rearranged such that xk xk+1 , k=1, . . . , n-1 x 1 A x 2 AR x 3 x 4 x 5 x 6 x 7 16

Principle of the ordinal regression n Example: Let AR={a 1, a 2, a 3}, G={Gain_1, Gain_2} Evaluation of reference actions on criteria Gain_1, Gain_2: Gain_1 Gain_2 a 1 4 6 a 2 5 5 a 3 6 4 Reference ranking: a 1 a 2 a 3 17

Principle of the ordinal regression 18

Principle of the ordinal regression 19

Principle of the ordinal regression n Let’s change the reference ranking: a 1 n a 2 a 3 a 2 One linear piece per each marginal value function u 1, u 2 is not enough Gain_1 Gain_2 a 1 4 6 a 2 5 5 a 3 6 4 u 1=k 1 Gain_1, u 2=k 2 Gain_2, U=u 1+u 2 For a 1 a 3, k 2>k 1, but for a 3 a 2, k 1>k 2, thus, marginal value functions cannot be linear 20

Principle of the ordinal regression 21

Principle of the UTA method n (Jacquet-Lagreze & Siskos, 1982) The comprehensive preference information is given in form of a complete preorder on a subset of reference actions AR A, AR={x 1, x 2, . . . , xn} – the reference actions are rearranged such that xk xk+1 , k=1, . . . , n-1 n The inferred value of each reference action x AR where + and - are potential errors of over- and under-estimation of the right value, resp. n The intervals [ i, i] are divided into i equal sub-intervals with the end points (i I) 22

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

UTA additive preference model 24

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

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

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

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

Współczynnik Kendalla n Następnie oblicza się współczynnik Kendalla : gdzie dk(R, R*) jest odległością Kendalla między macierzami R i R*: n Stąd -1, 1 n 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 n Jeżeli = 1, to zachodzi całkowita zgodność porządków z obydwu macierzy. W tej sytuacji błąd estymacji funkcji użyteczności F*=0 n W praktyce funkcję użyteczności akceptuje się, gdy 0. 75 29

Example of UTA+ n Ranking of 6 means of transportation 1 2 30

Preference attitude: „economical” 31

Preference attitude: „hurry” 32

Preference attitude: „hurry” 33

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

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

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

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

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

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

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

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

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

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

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

The UTAGMS method n 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 functions U verifying set E(x, y) of constraints: a b a a b b a, b AR E(x, y) a a a where is a small positive constant n For all (x, y) A, E(x, y) = E(y, x) 45

The UTAGMS method n N means necessary (strong) preference relation n Given a pair of actions x, y A x Ny d(x, y) 0 where n d(x, y) 0 means that for all compatible value functions x is at least as good as y n For x, y AR : x y x Ny 46

The UTAGMS method n P means possible (weak) preference relation n Given a pair of actions x, y A x Py D(x, y) 0 where n D(x, y) 0 means that for at least one compatible value function x is at least as good as y n For x, y AR : x y not y Px 47

The UTAGMS method n Some properties: n x Ny x Py n N is a partial preorder (i. e. N is reflexive and transitive) n P is strongly complete (i. e. for all x, y A, x Py or y Px) and negatively transitive (i. e. for all x, y, z A, not x Py and not y Pz not x Pz ), (in general, P is not transitive) n d(x, y) = Min{U(x)–U(y)} = –Max{–[U(x)–U(y)]} = = –Max{U(y)–U(x)} = –D(y, x) 48

Proof of transitivity of N n d(x, y)>0 means: Min{U(x)-U(y)}>0 n This is equivalent to the fact: for all value functions compatible with the reference preorder, U(x)>U(y) n The set of all compatible value functions is the same for calculation of d(x, y) for any pair x, y A n 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 n 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) n In other words, U*(x)>U*(y)>U*(z)>U*(x) n This is a contradiction, so N is transitive 49

The UTAGMS method n Elaboration of the final ranking: n 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 not x Ny and not y Nx n for the possible preference relation being complete ( P is supported by at least one compatible value function) n 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 are not dual 50

The UTAGMS method – final ranking (partial preorder) N N N n N N N Final ranking corresponding to necessary (strong) preference : 51

The UTAGMS method – final ranking (complete preorder) n Final ranking corresponding to possible (weak) preference : P P P P P 52

Example of UTAGMS n Ranking of 6 means of transportation 1 2 53

„hurry” D(x, y) 54

„hurry” „economical” 56

Nested ranking with different credibility levels n Reference rankings in growing sets: with credibility levels ordered decreasingly n The reference ranking of alternatives from ARi does not change in ARi+1, i=1, …, p-1 n Each time we pass from ARi to ARi+1, we add to (E) new constraints concerning alternatives from {ARi+1 ARi} n If d(x, y)<0 in iteration i turns to d(x, y)>0 in iteration i+1, then we assign to x Ny the credibility level corresponding to ranking i+1 n In this way we get a set of nested partial preorders n In fact, we get a fuzzy partial preorder N~ with respect to credibility: for all x, y, z A, Min{Cr(x N~y), Cr(y N~z)} Cr(x N~z), thus N~ is min-transitive and reflexive 57

Software demonstration Bank 58

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New reference action 60

New reference ranking 61

High credibility ranking embedded in low credibility ranking Partial preorder graph (strong outranking) 62

GRIP – Generalized Regression with Intensities of Preference (Figueira, Greco, Słowiński 2006) n GRIP extends the UTAGMS method by adopting all features of UTAGMS and by taking into account additional preference information : n 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” n 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” 63

GRIP – Generalized Regression with Intensities of Preference n DM is supposed to provide the following preference information : n a partial preorder on AR, such that x, y AR x y „x is at least as good as y” = not -1, n = -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” * = * not *-1, n * = * *-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* not i*-1, i* = i* i*-1 64

GRIP – new contraints of the ordinal regression LP problem n A value function U : [0, 1] is called compatible if it satisfes 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) ui(x) ui(y) iff x i y, i I h) ui(x) – ui(y) ui(w) – ui(z) iff (x, y) i* (w, z), i I i) ui(x) – ui(y) > ui(w) – ui(z) iff (x, y) i* (w, z), i I j) ui(x) – ui(y) = ui(w) – ui(z) iff (x, y) i* (w, z), i I 65

GRIP – new contraints of the ordinal regression LP problem n Moreover, the following normalization constraints should also be taken into account: k) ui( i)=0, i I l) 66

GRIP – new contraints of the ordinal regression LP problem n 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 : n for all x, y A, a necessary weak preference relation, x N y: U(x) U(y) for all compatible value functions n for all x, y A, a possible weak preference relation, x P y: U(x) U(y) for at least one compatible value function n for all x, y, w, z A, a necessary relation of preference intensity, (x, y) *N (w, z): [U(x) – U(y)] – [U(w) – U(z)] 0 for all compatible value functions n for all x, y, w, z A, a possible relation of preference intensity, (x, y) *P (w, z): [U(x) – U(y)] – [U(w) – U(z)] 0 for at least one compatible value function 67

GRIP – fundamental properties of N, P, *N, *P Theorem: If constaraints a) – l) are satisfied, then the properties hold: 1. For all x, y A, x N y x P y 2. For all x, y AR, x y x N y 3. N is a partial preorder (i. e. the relation is transitive and reflexive) and P is strongly complete and negatively transitive 4. For all x, y, z A, [x N y and y P z] x P z 5. For all x, y, z A, [x P y and y N z] x P z 6. For all x, y, w, z A, (x, y) *N (w, z) (x, y) *P (w, z) 7. For all x, y, w, z A, (x, y) * (w, z) (x, y) *N (w, z) 8. *N is a partial preorder and *P is strongly complete and negatively transitive 9. For all x, y, w, z, r, s A, [(x, y) *N (w, z) and (w, z) *P (r, s)] (x', y) *P (r, s) 10. For all x, y, w, z, r, s A, [(x, y) *P (w, z) and (w, z) *N (r, s)] (x, y) *P (r, s) 11. For all x, x’, y, w, z A, [x’ N x and (x, y) *N (w, z)] (x’, y) *N (w, z) 12. For all x, x’, y, w, z A, [x’ N x and (x, y) *P (w, z)] (x’, y) *P (w, z) 13. For all x, x’, y, w, z A, [x’ P x and (x, y) *N (w, z)] (x’, y) *P (w, z) 68

GRIP – fundamental properties of N, P, *N, *P 14. For all x, y, y’, w, z A, [y N y’ and (x, y) *N (w, z)] (x, y’) *N (w, z) 15. For all x, y, y’, w, z A, [y N y’ and (x, y) *P (w, z)] (x, y’) *P (w, z) 16. For all x, y, y’, w, z A, [y P y’ and (x, y) *N (w, z)] (x, y’) *P (w, z) 17. For all x, y, w, w’, z A, [w N w’ and (x, y) *N (w, z)] (x, y) *N (w’, z) 18. For all x, y, w, w’, z A, [w N w’ and (x, y) *P (w, z)] (x, y) *P (w’, z) 19. For all x, y, w, w’, z A, [w P w’ and (x, y) *N (w, z)] (x, y) *P (w’, z) 20. For all x, y, w, z, z’ A, [z’ N z and (x, y) *N (w, z)] (x, y) *N (w, z’) 21. For all x, y, w, z, z’ A, [z’ N z and (x, y) *P (w, z)] (x, y) *P (w, z’) 22. For all x, y, w, z, z’ A, [z’ P z and (x, y) *N (w, z)] (x, y) *P (w, z’) 23. For all x, x’, y A, (x’, y) *N (x, y) x’ N x 24. For all x, x’, y A, (x’, y) *P (x, y) x’ P x 25. For all x, y, y’ A, (x, y) *N (x, y’) y’ N y 26. For all x, y, y’ A, (x, y) *P (x, y’) y’ P y 69

GRIP – the linear programming problem n In order to verify the truth or falsity of necessary and possible weak preference relations N, P and *N, *P, one can use LP n LP does not permit strict inequalities, such as b), e), i) They must be rewritten 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 >0 (small value) n In UTA and in UTAGMS the result is dependent on the value of n We want to make the result of GRIP independent of 70

GRIP – the linear programming problem n The following result will be useful (see e. g. Marichal & Roubens 2000): n Proposition: x is a solution of the linear system, # if there exists >0, such that In particular, a solution exists, iff the following LP Max has optimal solution (x*, *), where *>0. Then, x* is a solution of # 71

GRIP – the linear programming problem n According to the Proposition, if constraints b), e), i) are considered, in order to verify the truth or falsity of N and P , one should : Max subject to constraints a)–l), with b), e), i) written as b’), e’), i’) n If maximal *>0, the set of compatible value functions is not empty n Then, to verify the truth or falsity of x Py, for any x, y A, one should : Max subject to constraints a)–l), with b), e), i) written as b’), e’), i’) and U(x) U(y) n Maximal *>0 x Py n This means that there exists at least one compatible value function satisfying the hypothesis U(x) U(y) 72

GRIP – the linear programming problem n In order to verify the truth or falsity of x Ny, rather than to check directly that for each compatible value function U(x) U(y), we make sure that among the compatible value 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) + n Maximal *≤ 0 x Ny 73

GRIP – the linear programming problem n Analogously, if constraints b), e), i) are considered, 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) n n Maximal *>0 (x, y) *P(w, z) 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) + n Maximal *≤ 0 (x, y) *N(w, z) n The value of * is not meaningful – the result does not depend on it! 74

Comparison of GRIP and MACBETH (Bana e Costa & Vansnick 1994) GRIP • Ordinal comprehensive preference inf. on pairwise comparison of some Preference information reference actions: x y, x, y AR • Absolute qualitative judgement of intensity of preference for some pairs of reference actions – partial and/or comprehensive (e. g. v_weak, moderate, . . . , extreme intensity of preference for (x, y)) OR • Comparison of intensities of preference for some pairs of reference actions – partial and/or comprehens. : (x, y) i(w, z) and/or (x, y) (w, z) MACBETH • Ordinal preference inf. w. r. t. each criterion for all not equally attractive pairs of actions: x iy or y ix, x, y A • Definition of „neutral” and „good” level on original scales of criteria • Absolute qualitative judgement of differences of attractiveness for all not equally attractive pairs of actions w. r. t. each criterion, including „good” and „neutral” points (e. g. v_weak, moderate, . . . , extreme int. pref. for (x, y)) • Ordinal preference inf. for all not equally attractive criteria: gi gj or gj gi • Absolute qualitative judgement of differences of attractiveness for all not equally attractive pairs of criteria (e. g. v_weak, moderate, . . . , extreme intensity of preference for (gi, gj)) 75

Comparison of GRIP and MACBETH cont. Preference model and final results GRIP • Uses LP to identify a set of comprehensive additive value functions with interval scales, compatible with preference info. • Builds necessary and possible weak preference relations on set A: • N (partial preorder) • P (strongly complete) • Builds necessary and possible weak preference relations on set A A: • *N (partial preorder) • *P (strongly complete) MACBETH • Uses LP to build a single interval scale for each criterion, compatible with preference info. , and computes a numerical marginal value for each action on each criterion • Computes a weight for each criterion • Builds a weighted sum model on marginal values which is additive piecewise linear or discrete • Uses the model to set up a complete preorder on set A 76

Comparison of GRIP and MACBETH n (Bana e Costa & Vansnick 1994) Summary of crucial differences in the methodology: n GRIP is using comprehensive and partial preference information on some pairs of actions n MACBETH requires partial preference information on all pairs of actions n Information about partial intensity of preference is of the same nature in GRIP and MACBETH (equivalence classes of relation i* correspond to qualitative judgements of MACBETH), but in GRIP it may not be complete n GRIP represents „disaggregation-aggregation” approach n MACBETH uses „aggregation” approach – needs weights to aggregate scales on particular criteria n GRIP works with all compatible value functions, while MACBETH builds a single interval scale for each criterion, even if many such scales would be compatible with preference information 77

Other features of GRIP n GRIP can be used interactively: n In the absence of any preference information, N, *N boil down to weak dominance relation n Each pairwise comparison or each comparison of intensities of preference *, contributes to enrich N or *N n In the absence of any preference information, P, *P is a complete relation n Each pairwise comparison or each comparison of intensities of preference *, contributes to impoverish P or *P n For complete pairwise comparisons and comparisons of intensities: N = P and *N = *P n GRIP permits to make preference intensity dependent on the part of criterion scale in which a difference of performances takes place, e. g. (17. 000; 19. 000) price (27. 000; 30. 000) 78

GRIP – illustrative example n Car ranking problem n Criteria: Intensity of preference: 79

GRIP – illustrative example Price Speed Space Fuel_cons. Acceleration Skoda Opel Ford Citroen Seat VW n Performance matrix 80

GRIP – illustrative example n Preference information n Monotonicity must be respected for each criterion, e. g. if Speed(x) Speed(y), then value[Speed(x)] value[Speed(y)] n In the ordinal regression LP problem, monotonicity is expressed by g) n In Tables 2 -6, we skip preference labels and that result from simple monotonicity, i. e. gi(x)=gi(y) or gi(x) gi(y), respectively 81

GRIP – illustrative example n Partial information about preference intensity serves to define constraints h) and j) representing partial preorder i* 82

GRIP – illustrative example (F) Skoda Opel Ford Citroen n Seat Comprehensive information about preference intensity serves to VW define constraints a) and c) representing partial preorder * 83

GRIP – illustrative example n Calculation of D(x, y)=Max{U(x)–U(y)} n Since all values are 0, for all pairs (x, y) of cars there exist at least one compatible value function for which x is at least as good as y n Therefore, possible weak preference relation P A A, so P brings no interesting information because all weak preferences are possible 84

GRIP – illustrative example n Calculation of d(x, y)=Min{U(x)–U(y)} n As Min{U(x)–U(y)}=–Max{U(y)–U(y)}, then d(x, y)=–D(y, x) n All values 0 correspond to pairs (x, y) of cars for which all compatible value functions are in favor of x over y 85

GRIP – illustrative example n Graph of necessary weak preference relation N 86

Conclusions n In GRIP, preference information is given by the DM in terms of: n partial preorder in the set of reference actions n partial and comprehensive comparisons of intensities of preference between some pairs of reference actions, n The preference information is used within regression approach to build a complete set of compatible additive value functions n Considering all compatible value functions permits to find as result: n necessary w. pref. relation in A and in A A (partial preorder) N, *N n possible w. pref. relation in A and in A A (strongly complete) P, *P n Possible extensions: n preference information with gradual credibility 87