Ranking with Multiple reference Points Efficient Elicitation and - PowerPoint PPT Presentation

Ranking with Multiple reference Points Efficient Elicitation and Learning Procedure Khaled Belahcène 1 - Vincent Mousseau 1 - Wassila Ouerdane 1 - Marc Pirlot 2 - Olivier Sobrie 2 1 CentraleSupélec - Université Paris-Saclay 2 University of Mons - Faculty of engineering June 21, 2019 University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 1 / 29

1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 2 / 29

Introduction 1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 3 / 29

Introduction Ranking alternatives/objects Problem ◮ Ranking alternative/object by preference ◮ e.g. ranking of cars ≻ ≻ MCDA ranking methods/models ◮ UTilités Additives (UTA) ◮ ELimination and Choice Expressing REality (ELECTRE II) ◮ Ranking with Multiple reference Points (RMP) University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 4 / 29

Ranking with Multiple reference Points (RMP) 1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 5 / 29

Ranking with Multiple reference Points (RMP) Ranking with Multiple reference Points (RMP) I ◮ Equi-important A 8 criteria 8 ◮ Reference points 28 s. r 2 R = { r 1 , r 2 } 18 k A C 4 B 31 s. r 1 2 12 k A C C Brakes Comfort Price Acceleration A A A B ≻ C A A A University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 6 / 29

Ranking with Multiple reference Points (RMP) Ranking with Multiple reference Points (RMP) II ◮ Equi-important A 8 criteria 8 ◮ Reference points 28 s. r 2 R = { r 1 , r 2 } 18 k A C 4 B 31 s. r 1 2 12 k A C C Brakes Comfort Price Acceleration A A A B ≻ C A A A ≻ B A B A ≻ University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 7 / 29

Ranking with Multiple reference Points (RMP) Ranking with Multiple reference Points (RMP) III Some characteristics of RMP ◮ Model introduced by Antoine Rolland (Rolland, 2013) ◮ Transitivity ensured ≻ ≻ ≻ ◮ Safe regarding rank-reversal ≻ ≻ ≻ ◮ No need for commensurate scales University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 8 / 29

Inferring the parameters of an RMP model 1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 9 / 29

Inferring the parameters of an RMP model Inferring the parameters of an RMP model I RMP model Learning set ≻ A 8 ≻ ≻ 8 28 s. Algorithm r 2 ∼ 18 k A C 4 B ∼ 31 s. r 1 ≻ 2 12 k A C ... C Brakes Comfort Price Acceleration University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 10 / 29

Inferring the parameters of an RMP model Inferring the parameters of an RMP model II Existing algorithms ◮ MIP-based algorithms (Zheng et al., 2012; Liu, 2016) ◮ S-RMP model (RMP with additive weights) ◮ Mixed Integer Program ◮ Minimization of Kemeny distance (Kemeny, 1959) ◮ Metaheuristic algorithm (Liu et al., 2014; Liu, 2016) ◮ S-RMP model (RMP with additive weights) ◮ Evolutionnary algorithm ◮ Reasonable computing time Limitations of the existing algorithms ◮ Additive representation of criteria importance relation ◮ MIP only able to handle very limited datasets ◮ Metaheuristic cannot always restore a S-RMP model University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 11 / 29

MAX-SAT formulation for inferring an RMP Model 1 Introduction 2 Ranking with Multiple reference Points (RMP) 3 Inferring the parameters of an RMP model 4 MAX-SAT formulation for inferring an RMP Model 5 Experimental results 6 Conclusion and further research University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 12 / 29

MAX-SAT formulation for inferring an RMP Model SAT formulation for inferring an RMP Model I Boolean Satisfiability problem ◮ Boolean variables V ; ◮ Logical proposition about these variables f : { 0 , 1 } V → { 0 , 1 } ; ◮ SATisfiable if v ∗ exists such that f ( v ∗ ) = 1 ◮ f can be expressed as conjunction of clauses C : c ∈C c ; f = � ◮ Each clause c ∈ C is a disjunction of their variables or their negation : ∀ c ∈ C , ∃ c + , c − ∈ P ( V ) : c = � v ∈ c + v ∨ � v ∈ c − ¬ v ; ◮ NP-complete problem BUT efficient SAT algorithms exist SAT for learning an RMP model ◮ Expression of constraints as a SAT problem ◮ Limited to strict preferences ( a ≻ b ) University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 13 / 29

MAX-SAT formulation for inferring an RMP Model SAT formulation for inferring an RMP Model II ϕ := ϕ scales ∧ ϕ profiles ∧ ϕ order ∧ ϕ outranking ∧ ϕ lexicography ∧ ϕ preference 1. ϕ scales : Monotonicity of criteria scales � � ϕ scales := ( x i , h , k ∨ ¬ x i , h , k ′ ) i ∈ N k ′ < k ∈ X i ◮ x i , h , k : equal to 1 if value k above reference point r h on criterion i ◮ N : set of criteria indices ◮ X i : set of values on criterion i University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 14 / 29

MAX-SAT formulation for inferring an RMP Model SAT formulation for inferring an RMP Model II ϕ := ϕ scales ∧ ϕ profiles ∧ ϕ order ∧ ϕ outranking ∧ ϕ lexicography ∧ ϕ preference 2. ϕ profiles : Dominance of the profiles ϕ profiles := ϕ profiles 1 ∧ ϕ profiles 2 � � � ϕ profiles 1 := ( x i , h ′ , k ∨ ¬ x i , h , k ∨ ¬ d h , h ′ ) h � = h ′ ∈ H i ∈ N k ∈ X i � ϕ profiles 2 := ( d h , h ′ ∨ d h ′ , h ) h < h ′ ∈ H ◮ N : set of criteria indices ◮ X i : set of values on criterion i ◮ H : set of reference points indices ◮ d h , h ′ : equal to 1 if value if r h dominates r h ′ ◮ x i , h , k : equal to 1 if value k above reference point r h on criterion i University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 14 / 29

MAX-SAT formulation for inferring an RMP Model SAT formulation for inferring an RMP Model II ϕ := ϕ scales ∧ ϕ profiles ∧ ϕ order ∧ ϕ outranking ∧ ϕ lexicography ∧ ϕ preference 3. ϕ order : Order among criteria sets ϕ order := ϕ Pareto ∧ ϕ completeness ∧ ϕ transitivity � ϕ Pareto := ( y B , A ) A ⊆ B ∈P ( N ) � ϕ completeness := ( y A , B ∨ y B , A ) A , B ∈P ( N ) � ϕ transitivity := ( ¬ y A , B ∨ ¬ y B , C ∨ y A , C ) A , B , C ∈P ( N ) ◮ P ( N ) : set of possible criteria coalitions ◮ y A , B : equal to 1 if criteria coalition A is more important than criteria coalition B University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 14 / 29

MAX-SAT formulation for inferring an RMP Model SAT formulation for inferring an RMP Model II ϕ := ϕ scales ∧ ϕ profiles ∧ ϕ order ∧ ϕ outranking ∧ ϕ lexicography ∧ ϕ preference 4. ϕ outranking : Outranking relation between pairs ϕ outranking := ϕ outranking 1 ∧ ϕ outranking 2 ∧ ϕ outranking 3 � � � � � ϕ outranking 1 := ( i ∨ y A , B ∨ ¬ z j , h ) x i , h , p j i ∨ ¬ x i , h , n j A , B ∈P ( N ) j ∈ J h ∈ H i / ∈ A i ∈ B ◮ p j ≻ n j : pairwise comparison j ◮ J : set of pairwise comparisons indices ◮ P ( N ) : set of possible criteria coalitions ◮ H : set of reference points indices ◮ x i , h , k : equal to 1 if value k above reference point r h on criterion i ◮ y A , B : equal to 1 if criteria coalition A is more important than criteria coalition B ◮ z j : equals to 1 if criteria set on which p j above r h is more important than the criteria set on which n j is above r h University of Mons K. Belahcène - V. Mousseau - W. Ouerdane - M. Pirlot - O. Sobrie - June 21, 2019 14 / 29