Analyzing the correspondence between non-strict and strict outranking relations

Transcription

1 Analyzing the correspondence between non-strict and strict outranking relations Denis Bouyssou Marc Pirlot CNRS Paris, France FPMs Mons, Belgium ROADEF, Toulouse, 2010

2 2 Introduction Outranking relations Concordance / non-discordance alternative x is preferable to alternative y if concordance condition the coalition of attributes supporting this assertion is more important than the coalition of attributes opposing it non-discordance condition there is no attribute that strongly opposes this assertion ELECTRE preferable means at least as good as TACTIC preferable means strictly better than

3 5 Framework Definitions and notation Setting Classical conjoint measurement setting N = {1, 2,..., n}: set of attributes X = n i=1 X i with n 2: set of alternatives x = (x 1, x 2,..., x n ) X S reflexive binary relation on X interpreted as an at least as good as relation between alternatives

4 6 Definitions and notation Reflexive Concordance relations Definition of reflexive concordance relations Reflexive Concordance Relations (RCR) with S(x, y) = {i N : x i S i y i } and x S y S(x, y) S(y, x) S i : complete binary relation X i : binary relation between subsets of attributes having N for union that is increasing w.r.t. inclusion and such that N N A B, C A and B D C D S i : preference relation on attribute i N asymmetric part of S i: S a i symmetric part of S i: S s i : importance relation between coalitions of attributes because S i is complete, S(x, y) S(y, x) = N, for all x, y X

5 7 Definitions and notation Outranking relations Definition of reflexive outranking relations Reflexive Outranking Relations (ROR) x S y [S(x, y) S(y, x) and V (y, x) = ] with S(x, y) = {i N : x i S i y i } and V (y, x) = {i N : y i V i x i } S i : complete binary relation X i V i : binary relation on X i such that V i S a i : binary relation between subsets of attributes having N for union that is increasing w.r.t. inclusion A B, C A and B D C D V i : far better than relation on attribute i N

6 8 ELECTRE I Definitions and notation Examples ELECTRE I with: x S y s [0.5, 1]: concordance threshold i S(x,y) w i j N w j and V (y, x) = s S i : semi order (complete, Ferrers and semitransitive) V i Si a : strict semiorder (asymmetric, Ferrers and semitransitive)

7 10 Axiomatic analysis Model Conjoint measurement framework Model (M) x S y F (p 1 (x 1, y 1 ), p 2 (x 2, y 2 ),..., p n (x n, y n )) 0 with p i skew symmetric (p i (x i, y i ) = p i (y i, x i )) F nondecreasing in all its arguments F (0) 0 (M) Interpretation p i measures preference differences between levels on attribute i N F synthesizes these preference differences

8 11 Axiomatic analysis Axiomatic analysis Results Model (M) RCR ROR two conditions guaranteeing that preference differences on each attribute are well behaved axioms for model (M) two additional axioms guaranteeing that each p i takes at most three distinct values: +k i, 0, k i upper coarseness, lower coarseness axioms for model (M) two additional axioms guaranteeing that each p i takes at most five distinct values: +v i, +k i, 0, k i, v i upper coarseness, weak lower coarseness

9 13 Extensions Asymmetric outranking relations Asymmetric outranking relations same principles, except that preferable means strictly preferred instead of at least as good as TACTIC, Vansnick, 1986 w i > i P(x,y) x P y and W (y, x) = with: P(x, y) = {i N : x i P i y i } ε 0: threshold j P(y,x) w j + ε P i : strict semi order (asymmetric, Ferrers and semitransitive) W i P i : strict semiorder (asymmetric, Ferrers and semitransitive)

10 14 Extensions Asymmetric outranking relations Definition of asymmetric concordance relations Asymmetric Concordance relations (ACR) with P(x, y) = {i N : x i P i y i } and x P y P(x, y) P(y, x) P i : asymmetric binary relation X i : binary relation between disjoint subsets of attributes that is increasing w.r.t. inclusion and such that Not[ ] A B, C A and B D C D

11 15 Coduality Extensions Asymmetric outranking relations Coduality T is a binary relation on A a T cd b Not[ b T a ] RCR and ACR complete RCR and ACR correspond through coduality

12 16 Coduality Extensions Asymmetric outranking relations x P y P(x, y) P(y, x) P i cd = S i (S i is complete) A B Not[ N \ B N \ A ] Not[ y P x ] Not[ P(x, y) P(y, x) ] Not[ N \ S(y, x) N \ S(x, y) ] S(x, y) S(y, x) x S y S codual of P is complete conversely the codual of the completion of a RCR is an ACR

13 17 Extensions Asymmetric outranking relations Definition of asymmetric outranking relations Asymmetric Outranking Relations (AOR) x P y [P(x, y) P(y, x) and W (y, x) = ] with P(x, y) = {i N : x i P i y i } and and W (y, x) = {i N : y i W i x i } P i : asymmetric binary relation X i W i : binary relation on X i such that W i P i : binary relation between disjoint subsets of attributes that is increasing w.r.t. inclusion and such that Not[ ] A B, C A and B D C D

14 18 Results Extensions Asymmetric outranking relations Not[ y P x ] Not[ P(x, y) P(y, x) and W (x, y) = ] Not[ N \ S(y, x) N \ S(x, y) ] or W (x, y) S(x, y) S(y, x) or W (x, y) Outranking relations with bonus x S y S(x, y) S(y, x) or W (x, y) we have a characterization of AOR we have a characterization of outranking relations with bonus through coduality

15 Extensions Asymmetric part of a ROR Asymmetric part of a ROR x S a y [x S y and Not[ y S x ] [S(x, y) S(y, x) and V (y, x) = ] and [Not[ S(y, x) S(x, y) ] or V (x, y) ] x c y iff S(x, y) S(y, x) x c y and V (y, x) = x S a y or x c y, V (y, x) = and V (x, y). 19 double rôle of the veto relation: veto and bonus is used in some outranking methods (ELECTRE TRI) is not an AOR is not an ROR an axiomatic characterization is available

16 References Bouyssou, D., Pirlot, M. (2005) A characterization of concordance relations. European Journal of Operational Research, 167, , Bouyssou, D., Pirlot, M. (2008) An axiomatic approach to TACTIC. Cahier du LAMSADE 238, Bouyssou, D., Pirlot, M. (2007) Further results on concordance relations. European Journal of Operational Research, 181, , Bouyssou, D., Pirlot, M. (2009) An axiomatic analysis of concordance-discordance relations. European Journal of Operational Research, 199, 468-Ű477, 2009.